Rd Sharma XII Vol 2 2021 Solutions for Class 12 Commerce Maths Chapter 12 Probability are provided here with simple step-by-step explanations. These solutions for Probability are extremely popular among Class 12 Commerce students for Maths Probability Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma XII Vol 2 2021 Book of Class 12 Commerce Maths Chapter 12 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma XII Vol 2 2021 Solutions. All Rd Sharma XII Vol 2 2021 Solutions for class Class 12 Commerce Maths are prepared by experts and are 100% accurate.

#### Question 1:

If one ball is drawn at random from each of three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, then the probability that 2 white and 1 black balls will be drawn is
(a) $\frac{13}{32}$

(b) $\frac{1}{4}$

(c) $\frac{1}{32}$

(d) $\frac{3}{16}$

#### Question 2:

A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is
(a) $\frac{44}{85×49}$

(b) $\frac{11}{85×49}$

(c) $\frac{13×24}{17×25×49}$

(d) none of these

#### Question 3:

A and B are two events such that P (A) = 0.25 and P (B) = 0.50. The probability of both happening together is 0.14. The probability of both A and B not happening is
(a) 0.39
(b) 0.25
(c) 0.11
(d) none of these

#### Question 4:

The probabilities of a student getting I, II and III division in an examination are $\frac{1}{10},\frac{3}{5}\mathrm{and}\frac{1}{4}$ respectively. The probability that the student fails in the examination is
(a) $\frac{197}{200}$

(b) $\frac{27}{100}$

(c) $\frac{83}{100}$

(d) none of these

#### Question 5:

India play two matches each with West Indies and Australia. In any match the probabilities of India getting 0,1 and 2 points are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points is
(a) 0.0875
(b) 1/16
(c) 0.1125
(d) none of these

#### Question 6:

Three faces of an ordinary dice are yellow, two faces are red and one face is blue. The dice is rolled 3 times. The probability that yellow red and blue face appear in the first second and third throws respectively, is
(a) $\frac{1}{36}$

(b) $\frac{1}{6}$

(c) $\frac{1}{30}$

(d) none of these

#### Question 7:

The probability that a leap year will have 53 Fridays or 53 Saturdays is
(a) $\frac{2}{7}$

(b) $\frac{3}{7}$

(c) $\frac{4}{7}$

(d) $\frac{1}{7}$

A leap year has 366 days

For a non-leap year:
52 weeks + 1 day

For a  leap year:
52 weeks + 2 days

#### Question 8:

A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is
(a) $\frac{1}{4}$

(b) $\frac{11}{24}$

(c) $\frac{15}{24}$

(d) $\frac{23}{24}$

4 letters can be placed in 4 envelopes in 4! ways = 24 ways
Now, there is only one method, by which all the letters are placed in the right envelope.

P(all letters are placed in the right envelopes) = $\frac{1}{24}$

P(all letters are not placed in the right envelopes) = 1 $-$ P(all letters are placed in the right envelopes)

$=1-\frac{1}{24}\phantom{\rule{0ex}{0ex}}=\frac{23}{24}$

#### Question 9:

A speaks truth in 75% cases and B speaks truth in 80% cases. Probability that they contradict each other in a statement, is
(a) $\frac{7}{20}$

(b) $\frac{13}{20}$

(c) $\frac{3}{5}$

(d) $\frac{2}{5}$

#### Question 10:

Three integers are chosen at random from the first 20 integers. The probability that their product is even is
(a) $\frac{2}{19}$

(b) $\frac{3}{29}$

(c) $\frac{17}{19}$

(d) $\frac{4}{19}$

#### Question 11:

Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is
(a) $\frac{14}{29}$

(b) $\frac{16}{29}$

(c) $\frac{15}{29}$

(d) $\frac{10}{29}$

(c) $\frac{15}{29}$

For sum of two integers to be odd, one integer should be even and the other should be odd.
In 30 consecutive integers, 15 are even and 15 are odd.

P(sum is odd) = P(first integer is odd and second is even) + P(first integer is even and second integer is odd)

$=\frac{15}{30}×\frac{15}{29}+\frac{15}{30}×\frac{15}{29}\phantom{\rule{0ex}{0ex}}=\frac{450}{30×29}\phantom{\rule{0ex}{0ex}}=\frac{15}{29}$

#### Question 12:

A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected randomwise, the probability that it is black or red ball is
(a) $\frac{1}{3}$

(b) $\frac{1}{4}$

(c) $\frac{5}{12}$

(d) $\frac{2}{3}$

#### Question 13:

Two dice are thrown simultaneously. The probability of getting a pair of aces is
(a) $\frac{1}{36}$

(b) $\frac{1}{3}$

(c) $\frac{1}{6}$

(d) none of these

#### Question 14:

An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at random. The probability that they are of the same colour is
(a) $\frac{5}{84}$

(b) $\frac{3}{9}$

(c) $\frac{3}{7}$

(d) $\frac{7}{17}$

#### Question 15:

A coin is tossed three times. If events A and B are defined as A = Two heads come, B = Last should be head. Then, A and B are
(a) independent
(b) dependent
(c) both
(d) mutually exclusive

#### Question 16:

Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 persons leaving at different floors is
(a) $\frac{{}^{7}P_{5}}{{7}^{5}}$

(b) $\frac{{7}^{5}}{{}^{7}P_{5}}$

(c) $\frac{6}{{}^{6}P_{5}}$

(d) $\frac{{}^{5}P_{5}}{{5}^{5}}$

#### Question 17:

A box contains 10 good articles and 6 with defects. One item is drawn at random. The probability that it is either good or has a defect is
(a) $\frac{64}{64}$

(b) $\frac{49}{64}$

(c) $\frac{40}{64}$

(d) $\frac{24}{64}$

#### Question 18:

A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a nail is

(a) $\frac{3}{16}$

(b) $\frac{5}{16}$

(c) $\frac{11}{16}$

(d) $\frac{14}{16}$

(c) $\frac{11}{16}$

#### Question 19:

A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that these are of the same colour is
(a) $\frac{5}{108}$

(b) $\frac{18}{108}$

(c) $\frac{30}{108}$

(d) $\frac{48}{108}$

#### Question 20:

If S is the sample space and P (A) = $\frac{1}{3}$ P (B) and S = AB, where A and B are two mutually exclusive events, then P (A) =
(a) 1/4
(b) 1/2
(c) 3/4
(d) 3/8

#### Question 21:

If A and B are two events, then P ($\overline{)A}$B) =
(a) P $\left(\overline{)A}\right)$ P $\left(\overline{)B}\right)$
(b) 1 − P (A) − P (B)
(c) P (A) + P (B) − P (AB)
(d) P (B) − P (AB)

(d) P (B) − P (AB)

#### Question 22:

If P (AB) = 0.8 and P (AB) = 0.3, then P $\left(\overline{)A}\right)$ + P $\left(\overline{)B}\right)$ =
(a) 0.3
(b) 0.5
(c) 0.7
(d) 0.9

#### Question 23:

A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then, the probability chosen to be white is
(a) 2/15
(b) 7/15
(c) 8/15
(d) 14/15

(c) 8/15

A white ball can be drawn in two mutually exclusive ways:
(I) Selecting bag X and then drawing a white ball from it.
(II) Selecting bag Y and then drawing a white ball from it.

Let E1, E2 and A be three events as defined below:
E1 = Selecting bag X
E2 = Selecting bag Y
A = Drawing a white ball

We know that one bag is selected randomly.

#### Question 24:

Two persons A and B take turns in throwing a pair of dice. The first person to throw 9 from both dice will be awarded the prize. If A throws first, then the probability that B wins the game is
(a) 9/17
(b) 8/17
(c) 8/9
(d) 1/9

#### Question 25:

The probability that in a year of 22nd century chosen at random, there will be 53 Sunday, is
(a) 3/28
(b) 2/28
(c) 7/28
(d) 5/28

​P(53 Sundays in a leap year) = $\frac{2}{7}\phantom{\rule{0ex}{0ex}}$

P(53 Sundays in a non-leap year) = $\frac{1}{7}$

There will be 24 leap years in the 22nd century, i.e. from the year 2201 to 2200, we will have 24 leap years.

∴ P(leap year) = $\frac{24}{100}$

P(non-leap year) = $\frac{76}{100}$

Now,

P(53 Sundays) = P(leap year)$×$P(53 Sundays in a leap year)
+ P(non-leap year)$×$P(53 Sundays in a non-leap year)

$=\frac{24}{100}×\frac{2}{7}+\frac{76}{100}×\frac{1}{7}\phantom{\rule{0ex}{0ex}}=\frac{48}{700}+\frac{76}{700}\phantom{\rule{0ex}{0ex}}=\frac{124}{700}\phantom{\rule{0ex}{0ex}}=\frac{31}{175}$

Disclaimer: None of the given options is correct.

#### Question 26:

From a set of 100 cards numbered 1 to 100, one card is drawn at random. The probability that the number obtained on the card is divisible by 6 or 8 but not by 24 is
(a) 6/25
(b) 1/4
(c) 1/6
(d) 2/5

Number divisible by 6 between 1 to 100 = 16
Number divisible by 8 between 1 to 100 = 12
Number divisible by 6 and 8 between 1 to 100 = 4
Number divisible by 24 between 1 to 100 = 4

P(number divisible by 6 or 8) = P(number divisible by 6) + P(number divisible by 8) $-$ P(number divisible by 6 and 8)

$=\frac{16}{100}+\frac{12}{100}-\frac{4}{100}\phantom{\rule{0ex}{0ex}}=\frac{24}{100}\phantom{\rule{0ex}{0ex}}=\frac{6}{25}$

P(number divisible by 6 or 8 but not by 24) = P(number divisible by 6 or 8) $-$ P(number divisible by 24)

$=\frac{6}{25}-\frac{4}{100}\phantom{\rule{0ex}{0ex}}=\frac{6}{25}-\frac{1}{25}\phantom{\rule{0ex}{0ex}}=\frac{5}{25}\phantom{\rule{0ex}{0ex}}=\frac{1}{5}$

Disclaimer: None of the given options is correct.

#### Question 27:

Mark the correct alternative in the following question:

If A and B are two events such that P(A) = $\frac{4}{5}$, and $\mathrm{P}\left(A\cap B\right)=\frac{7}{10}$, then P(B|A) =

Hence, the correct alternative is option (c).

#### Question 28:

Choose the correct alternative in the following question:

If A and B are two events associated to a random experiment such that , then P(A|B) =

Hence, the correct alternative is option (a).

#### Question 29:

Choose the correct alternative in the following question:

Associated to a random experiment two events A and B are such that . The value of P(A) is

Hence, the correct alternative is option (b).

#### Question 30:

Choose the correct alternative in the following question:

Hence, the correct alternative is option (b).

Disclaimer: The option (b) given in the book is incorrect as the probability of any event is always less than 1. The same has been corrected here.

#### Question 31:

Choose the correct alternative in the following question:

Hence, the correct alternative is option (a).

#### Question 32:

Choose the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 33:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 34:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 35:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 36:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 37:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 38:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 39:

If A and B are two events such that A ≠ Φ, B = Φ, then

(a) $P\left(A}{B}\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}$             (b)
(c) $P\left(A}{B}\right)=P\left(B}{A}\right)=1$           (d) $P\left(A}{B}\right)=\frac{P\left(A\right)}{P\left(B\right)}$

By the definition of conditional probability:
If A and B are two events such that A ≠ Φ, B = Φ, then $P\left(A}{B}\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}$.
Hence, the correct answer is option (a).

#### Question 40:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 41:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 42:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 43:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 44:

Mark the correct alternative in the following question:

A flash light has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then the probability that both are dead is

Hence, the correct alternative is option (a).

#### Question 45:

Mark the correct alternative in the following question:

A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability of getting exactly one red ball is

Hence, the correct alternative is option (b).

#### Question 46:

Mark the correct alternative in the following question:

A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability that exactly two of the three balls were red, the first ball being red, is

Hence, the correct alternative is option (b).

#### Question 47:

Mark the correct alternative in the following question:

In a college 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she failed in Mathematics is

Hence, the correct alternative is option (c).

#### Question 48:

Mark the correct alternative in the following question

Three persons, A, B and C fire a target in turn starting with A. Their probabilities of hitting the target are 0.4, 0.2 and 0.2, respectively. The probability of two hits is

(a) 0.024                                             (b) 0.452                                             (c) 0.336                                             (d) 0.188

Hence, the correct alternative is option (d).

Disclaimer: The option (d) in the textbook is incorrect. It should be 0.188 instead 0.138. The same has been corrected here.

#### Question 49:

A and B are two students. Their chances of solving a problem correctly are $\frac{1}{3}$ and $\frac{1}{4}$ respectively. If the probability of their making common error is $\frac{1}{20}$ and they obtain the same answer, then the probability of their answer to be correct is
(a) $\frac{10}{13}$            (b) $\frac{13}{120}$             (c) $\frac{1}{40}$             (d) $\frac{1}{12}$

E1 = they solve correctly.
E2 = they solve incorrectly.
A = they obtain the same result.

Hence, the correct answer is option (a).

#### Question 50:

Mark the correct alternative in the following question:

Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability that both cards are queen is

Hence, the correct alternative is option (a).

#### Question 51:

Mark the correct alternative in the following question:

A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is

Hence, the correct alternative is option (d).

#### Question 52:

Mark the correct alternative in the following question:

If two events are independent, then

(a) they must be mutually exclusive
(b) the sum of their probabilities must be equal to 1
(c) (a) and (b) both are correct
(d) none of the above is correct

Hence, the correct alternative is option (d).

#### Question 53:

Mark the correct alternative in the following question:

Two dice are thrown. If it is known that the sum of the numbers on the dice was less than 6, then the probability of getting a sum 3, is

Hence, the correct alternative is option (c).

#### Question 54:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (b).

#### Question 55:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 56:

Mark the correct alternative in the following question:

A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number of the die and a spade card is

Hence, the correct alternative is option (c).

#### Question 57:

Mark the correct alternative in the following question:

Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is

Hence, the correct alternative is option (d).

#### Question 58:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 59:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

Disclaimer: The answer given in the book is incorrect. The same has been corrected here.

#### Question 60:

If A and B are independent events such that 0 < P(A) < 1 and 0 < P(B) < 1., then which of the following is not correct?
(a) A and B are mutually exclusive
(b) A and $\overline{)B}$ are independent
(c) $\overline{)A}$ and B are independent
(d)  are independent

If A and B are independent events then P(AB) = P(A) P(B) ≠ 0
Since 0 < P(A) < 1 and 0 < P(B) < 1
A and B can never be mutually exclusive and A and $\overline{)B}$ are also independent
Similarly  , Also $\overline{)A}$ and B are independent

Hence, the correct answer is option A.

#### Question 61:

A card is picked at random from a pack of 52 playing cards. Given that the picked card is a queen, the probability of this card to be a card of spade is
(a) $\frac{1}{3}$
(b) $\frac{4}{13}$
(c) $\frac{1}{4}$
(d) $\frac{1}{2}$

Given that the picked card is a queen.
Therefore,

Since only one queen exists of spade.
Therefore,

Hence, the correct answer is option (c).

#### Question 62:

A die is thrown once. Let A be the event that the number obtained is greater than 3. let B be the event that the number obtained is less than 5. Then, P$\left(A\cup B\right)$ is
(a) $\frac{2}{5}$
(b) $\frac{3}{5}$
(c) $0$
(d) $1$

A: Number obtained is greater than 3
A = {4, 5, 6}
$n\left(A\right)=3$
$P\left(A\right)=\frac{3}{6}$

B: Number obtained is less than 5
A = {1, 2, 3, 4}
$n\left(A\right)=4$
$P\left(B\right)=\frac{4}{6}$

⋂ B: Number obtained is greater than 3 and less than 5
⋂ B = $\left\{4\right\}$
$P\left(A\cap B\right)=\frac{1}{6}$

We know
$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\phantom{\rule{0ex}{0ex}}⇒P\left(A\cup B\right)=\frac{3}{6}+\frac{4}{6}-\frac{1}{6}=1$
Hence, the correct answer is option (d).

#### Question 1:

If A and $\overline{)B}$ are independent events, then $P\left(\overline{)A}\cup B\right)=1-x$ where x = _______________.

Given  A and $\overline{B}$ are independent events and $P\left(\overline{)A}\cup B\right)=1-x$

#### Question 2:

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A, B) = $\frac{5}{9}$, then p = _________.

Since A and B are independent
⇒ $\overline{A}$  and B are independent and A and $\overline{B}$ are independent
Given P(A) = p
i.e P$\left(\overline{A}\right)$ = 1 – p
P(B) = 2p
i.e P$\left(\overline{B}\right)$ = 1 – 2p and P(exactly one of A, B) = $P\left(\overline{A}\cap B\right)+P\left(A\cap \overline{B}\right)$

i.e $\frac{5}{9}$ = (1 – P) (2p) + P(1– 2p)
= 2p – 2p2 + p – 2p2
i.e $\frac{5}{9}$ = 3p – 4p2
i.e 5 = 27p – 36p2
i.e 36p2 – 27p + 5 = 0
i.e 36p2 – 27p + 5 = 0
i.e D = (27)2 – 4(36)(5)
i.e D = 729 – 720
i.e D = 9

#### Question 3:

If A and B are two events such that P  = ______________.

Given, for two events A and B

#### Question 4:

If A and B are two events such that P(A/B) = p, P(A) = p then p = _____________.

for events A and B, Given

#### Question 5:

Let A and B be two events. If P(A/B) = P(A), then A is ______________ of B.

For two events A and B.
If P(A/B) = P(A)
i.e occurrence of A does not depend on occurrence of B.

Hence, A is independent of B.

#### Question 6:

Let A and B be two events such that  then k = ______________.

For two events, given

#### Question 7:

If two events A and B are mutually exclusive, then P(A/B) = ____________.

for two mutually exclusive events A and B, $P\left(A\cap B\right)$ = 0

.

#### Question 8:

If A and B are two events such that $A\subseteq B,$ then P(B/A) = _____________.

For two events A and
Such that $A\subseteq B$.

#### Question 9:

If 4 P(A) =6 P(B) = 10 P$\left(A\cap B\right)$ = 1, then P(B/A) = ______________.

For any two events A and B, given 4 P(A) = 6 P(B) = 10 P$\left(A\cap B\right)$ = 1
i.e

#### Question 10:

If A and B are two events, then P$\left(\overline{)A}\cap B\right)$ = _____________.

For two events A and B
P$\left(\overline{)A}\cap B\right)$ = ??
Using Venn-diagram

#### Question 11:

If A and B are two events, then the probability of occurrence of A only is equal to _____________.

For two events A and B, probability of occurrence of A only is $P\left(A\cap \overline{)B}\right)$
By Venn-diagram

∴ Probability of occurrence of A only is P(A) – P(AB).

#### Question 12:

If A and B are two events, then the probability of occurrence of exactly one of A and B is equal to __________.

For two events A and B,

Probability of occurrence of exactly are of A and B is $P\left(A\cap \overline{)B}\right)+P\left(A\cap \overline{B}\right)$

From previous two questions,

i.e Probability of occurrence of exactly are of A and B is

#### Question 13:

For two event A and B, if P(A) = P(A/B) = , then A and B are _____________ events.

For two events A and B
P(A) = P(A/B) =

A and B are independent events.

#### Question 14:

Let A and B be two events for which P(A) = a, P(B) = b = __________________.

For two events A and B P(A) = a, P(B) = b$P\left(A\cap B\right)=c$

Since $P\left(\overline{)A}\cap B\right)=P\left(B\right)-P\left(A\cap B\right)$

i.e $P\left(\overline{)A}\cap B\right)=b-c$

#### Question 15:

Let A, B, C be pairwise independent events with P(C) > 0 and  then x = ____________.

For given sets, A, B and C with P(C) > 0, P(ABC) = 0 and A, B, C are pairwise independent.
By deffination $P\left(\frac{{A}^{c}\cap {B}^{c}}{C}\right)=P\frac{\left({A}^{c}\cap {B}^{c}\cap C\right)}{P\left(C\right)}$

x = P(A) + P(B)

#### Question 16:

Let A, B, C be three events such that $P\left(A\cap B\cap C\right)=0,$ P(Exactly one of A and B occurs) = x, P (exactly one of B and C occurs) = y, P(Exactly one of A and C occurs) = z. Then $P\left(A\cup B\cup C\right)$ = ____________.

For any three given events A, B, C ; P(ABC) = 0.

P(Exactly one of A and B occurs) = x
i.e

and P (exactly one of B and C occurs) = y
i.e

also, P(Exactly one of A and C occurs) = z
i.e

Since

#### Question 17:

If A and B are two events such that $P\left(A\cup B\right)=\left(A\cap B\right),$ then P (exactly one of A and B occurs) = _____________.

For two events,  $P\left(A\cup B\right)=\left(A\cap B\right),$ then P (exactly one of A and B occurs)

P(exactly are of A and B occurs) = 0

#### Question 1:

A four digit number is formed using the digits 1, 2, 3, 5 with no repetitions. Write the probability that the number is divisible by 5.

#### Question 2:

When three dice are thrown, write the probability of getting 4 or 5 on each of the dice simultaneously.

#### Question 3:

Three digit numbers are formed with the digits 0, 2, 4, 6 and 8. Write the probability of forming a three digit number with the same digits.

#### Question 4:

A ordinary cube has four plane faces, one face marked 2 and another face marked 3, find the probability of getting a total of 7 in 5 throws.

#### Question 5:

Three numbers are chosen from 1 to 20. Find the probability that they are consecutive.

#### Question 6:

6 boys and 6 girls sit in a row at random. Find the probability that all the girls sit together.

#### Question 7:

If A and B are two independent events such that P (A) = 0.3 and P (A$\overline{)B}$) = 0.8. Find P (B).

#### Question 8:

An unbiased die with face marked 1, 2, 3, 4, 5, 6 is rolled four times. Out of 4 face values obtained, find the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5.

#### Question 9:

If A and B are two events write the expression for the probability of occurrence of exactly one of two events.

#### Question 10:

Write the probability that a number selected at random from the set of first 100 natural numbers is a cube.

#### Question 11:

In a competition A, B and C are participating. The probability that A wins is twice that of B, the probability that B wins is twice that of C. Find the probability that A losses.

#### Question 12:

If A, B, C are mutually exclusive and exhaustive events associated to a random experiment, then write the value of P (A) + P (B) + P (C).

#### Question 13:

If two events A and B are such that P $\left(\overline{)A}\right)$ = 0.3, P (B) = 0.4 and P (A$\overline{)B}$) = 0.5, find P $\left(B/\overline{)A}\cap \overline{)B}\right)$.

Disclaimer: the question seems to be incorrect.

#### Question 14:

If A and B are two independent events, then write P (A$\overline{)B}$) in terms of P (A) and P (B).

#### Question 15:

If P (A) = 0.3, P (B) = 0.6, P (B/A) = 0.5, find P (AB).

#### Question 16:

If A, B and C are independent events such that P(A) = P(B) = P(C) = p, then find the probability of occurrence of at least two of A, B and C.

So, the probability of occurrence of at least two of AB and C $3{p}^{2}-2{p}^{3}$.

Disclaimer: The answer given in the book is incorrect. the same has been corrected above.

#### Question 17:

If A and B are independent events, then write expression for P(exactly one of AB occurs).

#### Question 18:

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A and B occurs) = $\frac{5}{9}$, then find the value of p.

#### Question 1:

Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Sample space, S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Consider the given events.
A = An even number on the card
B = A number more than 3 on the card

Clearly,
A = {2, 4, 6, 8, 10}
B = {4, 5, 6, 7, 8, 9, 10}

#### Question 2:

Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?

Consider the given events.
A = Both the children are girls.
B = The youngest child is a girl.
C = At least one child is a girl.

#### Question 3:

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

Consider the given events
A = Numbers appearing on two dice are different
B = The sum of the numbers on two dice is 4

Clearly,
A = {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 4), (3, 5), (3, 6)
(4, 1), (4, 2), (4, 3), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}
B = {(1, 3), (3, 1) and (2, 2)}

#### Question 4:

A coin is tossed three times, if head occurs on first two tosses, find the probability of getting head on third toss.

Consider the given events.
A = Getting head on third toss
B = Getting head on first two tosses

Clearly,
A = {(H, H, H), (H, T, H), (T, H, H), (T, T, H)}
B = {(H, H, H), (H, H, T)}

#### Question 5:

A die is thrown three times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectively on first two tosses.

Consider the given events.
A = Getting 4 on third throw
B = Getting 6 on first throw and 5 on second throw

Clearly,
A = {(1, 1, 4), (1, 2, 4), (1, 3, 4), (1, 4, 4), (1, 5, 4), (1, 6, 4),
(2, 1, 4), (2, 2, 4), (2, 3, 4), (2, 4, 4), (2, 5, 4), (2, 6, 4),
.                                                                                 .
.                                                                                 .
.                                                                                 .
(6, 1, 4), (6, 2, 4), (6, 3, 4), (6, 4, 4), (6, 5, 4), (6, 6, 4)}
B = {6, 5, 1), (6, 5, 2), (6, 5, 3), (6, 5, 4), (6, 5, 5), (6, 5, 6)}

#### Question 6:

Compute P (A/B), if P (B) = 0.5 and P (AB) = 0.32

#### Question 7:

If P (A) = 0.4, P (B) = 0.3 and P (B/A) = 0.5, find P (AB) and P (A/B).

$\mathrm{Given}:\phantom{\rule{0ex}{0ex}}P\left(A\right)=0.4\phantom{\rule{0ex}{0ex}}P\left(B\right)=0.3\phantom{\rule{0ex}{0ex}}P\left(B/A\right)=0.5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\phantom{\rule{0ex}{0ex}}P\left(B/A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}\phantom{\rule{0ex}{0ex}}⇒0.5=\frac{P\left(A\cap B\right)}{0.4}\phantom{\rule{0ex}{0ex}}⇒P\left(A\cap B\right)=0.2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}P\left(A/B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{0.2}{0.3}=\frac{2}{3}$

#### Question 8:

If A and B are two events such that P (A) = $\frac{1}{3},$ P (B) = $\frac{1}{5}$ and P (AB) = $\frac{11}{30}$, find P (A/B) and P (B/A).

#### Question 9:

A couple has two children. Find the probability that both the children are (i) males, if it is known that at least one of the children is male. (ii) females, if it is known that the elder child is a female.

Consider the given events.
A = Both the children are female.
B = The elder child is a female.
C = At least one child is a male.
D = Both children are male.

[Here, first child is elder and second is younger]

#### Question 1:

From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that both of them are kings.

Consider the given events.
A = A king in the first draw
B = A king in the second draw

#### Question 2:

From a pack of 52 cards, 4 are drawn one by one without replacement. Find the probability that all are aces(or kings).

Consider the given events.
A = An ace in the first draw
B = An ace in the second draw
C = An ace in the third draw
D = An ace in the fourth draw

In case of kings, the required probablity will be = $\frac{1}{270725}$

#### Question 3:

Find the chance of drawing 2 white balls in succession from a bag containing 5 red and 7 white balls, the ball first drawn not being replaced.

Consider the given events.
A = A white ball in the first draw
B = A white ball in the second draw

#### Question 4:

A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.

There are 12 even numbers between 1 to 25.

Consider the given events.
A = An even number ticket in the first draw
B = An even number ticket in the second draw

#### Question 5:

From a deck of cards, three cards are drawn on by one without replacement. Find the probability that each time it is a card of spade.

Consider the events
A = An ace in the first draw
B = An ace in the second draw
C = Getting an ace in the third draw

#### Question 6:

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that
(i) both are kings
(ii) the first is a king and the second is an ace
(iii) the first is a heart and second is red.

(i) Consider the given events
A = A king in the first draw
B = A king in the second draw

(ii) Consider the given events
A = A king in the first draw
B = An ace in the second draw

(iii) Consider the given events.
A = A heart in the first throw
B = A red card in the second throw

#### Question 7:

A bag contains 20 tickets, numbered from 1 to 20. Two tickets are drawn without replacement. What is the probability that the first ticket has an even number and the second an odd number.

There are 10 even numbers and 10 odd numbers between 1 to 20.

Consider the given events.
A = An even number in the first draw
B = An odd number in the second draw

#### Question 8:

An urn contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that at least one ball is black?

Consider the given events.
A = A white or red ball in the first draw
B = A white or red ball in the second draw

#### Question 9:

A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement, find the probability that none is red.

Consider the given events.
A = A white or black ball in the first draw
B = A white or black ball in the second draw
C = A white or black ball in the third draw

#### Question 10:

A card is drawn from a well-shuffled deck of 52 cards and then a second card is drawn. Find the probability that the first card is a heart and the second card is a diamond if the first card is not replaced.

Consider the given events.
A = A heart in the first draw
B = A diamond in the second draw

#### Question 11:

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

Consider the given events.
A = A black ball in the first draw
B = A black ball in the second draw

#### Question 12:

Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and third card drawn is an ace?

Consider the given events.
A = A king in the first draw
B = A king in the second draw
C = An ace in the third draw

#### Question 13:

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

Consider the given events.
A = A good orange in the first draw
B = A good orange in the second draw
C = A good orange in the third draw

#### Question 14:

A bag contains 4 white, 7 black and 5 red balls. Three balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively.

Consider the given events.
A = A white ball in the first draw
B = A black ball in the second draw
C = A red ball in the third draw

#### Question 1:

If P (A) = $\frac{7}{13}$, P (B) = $\frac{9}{13}$ and P (AB) = $\frac{4}{13}$, find P (A/B).

#### Question 2:

If A and B are events such that P (A) = 0.6, P (B) = 0.3 and P (AB) = 0.2, find P (A/B) and P (B/A).

#### Question 3:

If A and B are two events such that P (AB) = 0.32 and P (B) = 0.5, find P (A/B).

#### Question 4:

If P (A) = 0.4, P (B) = 0.8, P (B/A) = 0.6. Find P (A/B) and P (AB).

#### Question 5:

If A and B are two events such that

[NCERT EXEMPLAR]

#### Question 6:

If A and B are two events such that 2 P (A) = P (B) = $\frac{5}{13}$ and P (A/B) = $\frac{2}{5},$ find P (AB).

#### Question 7:

If P (A) = $\frac{6}{11},$ P (B) = $\frac{5}{11}$ and P (AB) = $\frac{7}{11},$ find
(i) P (AB)
(ii) P (A/B)
(iii) P (B/A)

#### Question 8:

A coin is tossed three times. Find P (A/B) in each of the following:
(i) A = Heads on third toss, B = Heads on first two tosses
(ii) A = At least two heads, B = At most two heads
(iii) A = At most two tails, B = At least one tail.

(i) Consider the given events.
A = Heads on third toss
B = Heads on first two tosses

Clearly,
A = {(H, H, H), (H, T, H), (T, H, H), (T, T, H)}
B = {(H, H, H), (H, H, T)}

(ii) Consider the given events.
A = At least two heads
B = At most two heads

Clearly,
A = {(H, H, H), (H, T, H), (T, H, H), (H, H, T)}
B = {(T, T, T), (H, T, H), (T, H, H), (H, H, T), (T, H, T), (H, H, T), (H, T, T)}

(iii) Consider the given events.
A = At most two tails
B = At least one tail

Clearly,
A = {(T, T, H), (T, H, H), (H, H, T), (T, H, T), (H, H, T), (H, T, T), (H, H, H)}
B = {(T, T, T), (T, T, H), (T, H, H), (H, H, T), (T, H, T), (H, H, T), (H, T, T)}

#### Question 9:

Two coins are tossed once. Find P (A/B) in each of the following:
(i) A = Tail appears on one coin, B = One coin shows head.
(ii) A = No tail appears, B = No head appears.

(i) Consider the given events.
A = Tail appears on one coin
B = One coin shows head

Clearly,
A = {(H, T), (T, H)}
B = {(H, T), (T, H)}

(ii) Consider the given events.
A = No tail appears

Clearly,
A = {(H, H)}
B = {(T, T)}

#### Question 10:

A die is thrown three times. Find P (A/B) and P (B/A), if
A = 4 appears on the third toss, B = 6 and 5 appear respectively on first two tosses.

Consider the given events.
A = Getting 4 on third throw
B = Getting 6 on first throw and and 5 on second throw

Clearly,
A = {(1, 1, 4), (1, 2, 4), (1, 3, 4), (1, 4, 4), (1, 5, 4), (1, 6, 4), (2, 1, 4), (2, 2, 4), (2, 3, 4), (2, 4, 4), (2, 5, 4), (2, 6, 4),
.                                                                                 .
.                                                                                 .
.                                                                                 .
(6, 1, 4), (6, 2, 4), (6, 3, 4), (6, 4, 4), (6, 5, 4), (6, 6, 4)}
B = {6, 5, 1), (6, 5, 2), (6, 5, 3), (6, 5, 4), (6, 5, 5), (6, 5, 6)}

#### Question 11:

Mother, father and son line up at random for a family picture. If A and B are two events given by A = Son on one end, B = Father in the middle, find P (A/B) and P (B/A).

Consider the given events.
A = Son standing on one end
B = Father standing in the middle

#### Question 12:

A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

Consider the given events.
A = 4 appears on the die
B = The sum of the numbers on two dice is 6.

Clearly,
A = {(1, 4) (2, 4), (3, 4),(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 4), (6, 4)}
B = {(1, 5), (5, 1), (2, 4), (4, 2),(3, 3)}

#### Question 13:

Two dice are thrown. Find the probability that the numbers appeared has the sum 8, if it is known that the second die always exhibits 4.

Consider the given events.
A = 4 appears on second die
B = The sum of the numbers on two dice is 8.

Clearly,
A = {(1, 4), (2, 4), (3, 4), (4, 4) (5, 4) (6, 4)}
B = {(4, 4), (3, 5), (5, 3) (2, 6), (6, 2)}

#### Question 14:

A pair of dice is thrown. Find the probability of getting 7 as the sum, if it is known that the second die always exhibits an odd number.

Consider the given events.
A = Number appearing on second die is odd
B = The sum of the numbers on two dice is 7.

Clearly,
A = {(1, 1), (1, 3),(1, 5), (2, 1), (2, 3), (2, 5), (3, 1), (3, 3), (3, 5), (4, 1), (4, 3), (4, 5), (5, 1), (5, 3), (5, 5),(6, 1), (6, 3), (6, 5)}
B = {(2, 5), (5, 2), (3, 4), (4, 3), (1, 6), (6, 1)}

#### Question 15:

A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.

Consider the given events
A = A prime number appears on second die.
B = The sum of the numbers on two dice is 7.

Clearly,
A = {(1, 2), (1, 3), (1, 5), (2, 2), (2, 3)(2, 5), (3, 2), (3, 3), (3, 5) (4, 2), (4, 3), (4, 5),(5, 2), (5, 3), (5, 5), (6, 2), (6, 3),(6, 5)}
B = {(2, 5), (5, 2), (3, 4), (4, 3), (1, 6), (6, 1)}

#### Question 16:

A die is rolled. If the outcome is an odd number, what is the probability that it is prime?

Consider the given events.
A = The number is odd
B = The number is prime

Clearly,
A = {1, 3, 5}
B = {2, 3,5}

#### Question 17:

A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.

Consider the given events.
A = 4 appears on first die
B = The sum of the numbers on two dice is 8 or more.

Clearly,
A = {(4, 1), (4, 2), (4, 3), (4, 4) (4, 5), (4, 6)}
n(A) = 6

B = {(2, 6), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6),(5, 3), (5, 4), (5, 5) (5, 6), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
n(B) = 15

#### Question 18:

Find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show five.

Consider the given events.
A = At least one die does not show 5
B = The sum of the numbers on two dice is 8.

Clearly,
A = {(1, 1), (1, 2) (1, 3), (1, 4),(1, 6),(2, 1), (2, 2) (2, 3), (2, 4), (2, 6), (3, 1), (3, 2), (3, 3) (3, 4), (3, 6),(4, 1), (4, 2), (4, 3), (4, 4), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 6)}
B = {(2, 6), (3, 5), (4, 4), (5, 3),(6, 2)}

#### Question 19:

Two numbers are selected at random from integers 1 through 9. If the sum is even, find the probability that both the numbers are odd.

#### Question 20:

A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?

Consider the given events.
A = 5 appears on the die at least once
B = The sum of the numbers on two dice is 8.

Clearly,
A = {(1, 5),(2, 5),(3, 5),(4, 5),(5, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6)}
B = {(2, 6), (3, 5), (4, 4), (5, 3),(6, 2)}

#### Question 21:

Two dice are thrown and it is known that the first die shows a 6. Find the probability that the sum of the numbers showing on two dice is 7.

Consider the given events.
A = First die shows 6
B = The sum of the numbers on two dice is 7.

Clearly,
A= {(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
B = {(2, 5), (5, 2), (4, 3), (3, 4), (1, 6), (6, 1)}

#### Question 22:

A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event "5 appears on the first-die". Find P (E/F). If F is the event "5 appears on at least one die", find P (E/F).

Consider the given events.
E = The sum of the numbers on two dice is 10 or more
F = 5 appears on first die

Clearly,
E = {(4, 6),(5, 5),(5, 6),(6, 4), (6, 5), (6, 6)}
F = {(5, 1), (5, 2), (5, 3), (5, 4) (5, 5), (5, 6)}

Second case:
Consider the given events.
E = The sum of the numbers on two dice is 10 or more
F = 5 appears on a die at least once

Clearly,
E = {(4, 6),(5, 5),(5, 6),(6, 4), (6, 5), (6, 6)}
F = {(1, 5),(2, 5),(3, 5),(4, 5),(5, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6)}

#### Question 23:

The probability that a student selected at random from a class will pass in Mathematics is $\frac{4}{5},$ and the probability that he/she passes in Mathematics and Computer Science is $\frac{1}{2}$. What is the probability that he/she will pass in Computer Science if it is known that he/she has passed in Mathematics?

Consider the given events.
M = Students passes Mathematics
C = Students passes Computer Science

#### Question 24:

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

#### Question 25:

In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in class XII given that the chosen student is a girl?

#### Question 26:

Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Sample space, S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Consider the given events.
A = Even number appears on the card
B = A number, which is more than 3, appears on the card

Here,
A = {2, 4, 6, 8, 10}
B = {4, 5, 6, 7, 8, 9, 10}

#### Question 27:

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the constitutional probability that both are girls? Given that

(i) the youngest is a girl                                                 (b) at least one is a girl.                                                                                [CBSE 2014]

Let G and B represents respectively a girl and a boy.

As, the possible outcomes of a family having two children is given by the set, S = {GG, GB, BG, BB}, where first letter represents the elder child.

So, n(S) = 4

Now,

#### Question 1:

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
(i) A = the first throw results in head, B = the last throw results in tail
(ii) A = the number of heads is odd, B = the number of tails is odd
(iii) A = the number of heads is two, B = the last throw results in head

#### Question 2:

Prove that in throwing a pair of dice, the occurrence of the number 4 on the first die is independent of the occurrence of 5 on the second die.

#### Question 3:

A card is drawn from a pack of 52 cards so the teach card is equally likely to be selected. In which of the following cases are the events A and B independent?
(i) A = The card drawn is a king or queen, B = the card drawn is a queen or jack
(ii) A = the card drawn is black, B = the card drawn is a king
(iii) B = the card drawn is a spade, B = the card drawn in an ace

#### Question 4:

A coin is tossed three times. Let the events A, B and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row.
Check the independence of (i) A and B (ii) B and C and (iii) C and A

#### Question 5:

If A and B be two events such that P (A) = 1/4, P (B) = 1/3 and P (AB) = 1/2, show that A and B are independent events.

#### Question 6:

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find
(i) P (AB)
(ii) P (A$\overline{)B}$)
(iii) P ($\overline{)A}$B)
(iv)
(v) P (AB)
(vi) P (A/B)
(vii) P (B/A)

#### Question 7:

If P (not B) = 0.65, P (AB) = 0.85, and A and B are independent events, then find P (A).

$P\left(\overline{B}\right)=0.65\phantom{\rule{0ex}{0ex}}⇒1-P\left(B\right)=0.65\phantom{\rule{0ex}{0ex}}⇒P\left(B\right)=1-0.65=0.35\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\phantom{\rule{0ex}{0ex}}P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\phantom{\rule{0ex}{0ex}}⇒P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\right)×P\left(B\right)\phantom{\rule{0ex}{0ex}}⇒0.85=P\left(A\right)+0.35-0.35×P\left(A\right)\phantom{\rule{0ex}{0ex}}⇒0.85-0.35=P\left(A\right)\left[1-0.35\right]\phantom{\rule{0ex}{0ex}}⇒P\left(A\right)=\frac{0.5}{0.65}=0.77$

#### Question 8:

If A and B are two independent events such that P ($\overline{)A}$B) = 2/15 and P (A$\overline{)B}$) = 1/6, then find P (B).

#### Question 9:

A and B are two independent events. The probability that A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of occurrence of two events.

#### Question 10:

If A and B are two independent events such that P (AB) = 0.60 and P (A) = 0.2, find P (B).

#### Question 11:

A die is tossed twice. Find the probability of getting a number greater than 3 on each toss.

#### Question 12:

Given the probability that A can solve a problem is 2/3 and the probability that B can solve the same problem is 3/5. Find the probability that none of the two will be able to solve the problem.

#### Question 13:

An unbiased die is tossed twice. Find the probability of getting 4, 5, or 6 on the first toss and 1, 2, 3 or 4 on the second toss.

#### Question 14:

A bag contains 3 red and 2 black balls. One ball is drawn from it at random. Its colour is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing (i) two red balls, (ii) two black balls, (iii) first red and second black ball.

#### Question 15:

Three cards are drawn with replacement from a well shuffled pack of cards. Find the probability that the cards drawn are king, queen and jack.

#### Question 16:

An article manufactured by a company consists of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not be defective.

#### Question 17:

The probability that A hits a target is 1/3 and the probability that B hits it, is 2/5, What is the probability that the target will be hit, if each one of A and B shoots at the target?

#### Question 18:

An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

#### Question 19:

The odds against a certain event are 5 to 2 and the odds in favour of another event, independent to the former are 6 to 5. Find the probability that (i) at least one of the events will occur, and (ii) none of the events will occur.

#### Question 20:

A die is thrown thrice. Find the probability of getting an odd number at least once.

#### Question 21:

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are red, (ii) first ball is black and second is red, (iii) one of them is black and other is red.

#### Question 22:

An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting (i) 2 red balls, (ii) 2 blue balls, (iii) one red and one blue ball.

#### Question 23:

The probabilities of two students A and B coming to the school in time are $\frac{3}{7}\mathrm{and}\frac{5}{7}$ respectively. Assuming that the events, 'A coming in time' and 'B coming in time' are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.

#### Question 24:

Two dice are thrown together and the total score is noted. The event E, F and G are "a total of 4", "a total of 9 or more", and "a total divisible by 5", respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.               [NCERT EXEMPLAR]

We have,

S =
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

n(S) = 36

E = "a total of 4" = {(1, 3), (2, 2), (3, 1)} i.e. n(E) = 3
F = "a total of 9 or more = {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)} i.e. n(F) = 10
G = "a total divisible by 5" = {(1, 4), (2, 3), (3, 2), (4, 1), (4, 6), (5, 5), (6, 4)} i.e. n(G) = 7

Now,

#### Question 25:

Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are:

(i) p1p2               (ii) (1 $-$ p1)p2               (iii) 1 $-$ (1 $-$ p1)(1 $-$ p2)               (iv) p1p2 $-$ 2p1p2                       [NCERT EXEMPLAR]

#### Question 26:

The probability of finding a green signal on a busy crossing X is 30%. What is the probability of finding a green signal on X in two consecutive days out of three?

Let G be the event of green signal.
$\therefore$Required probability = $\mathrm{P}\left({\mathrm{GGG}}^{1}\right)+\mathrm{P}\left({\mathrm{G}}^{1}\mathrm{GG}\right)$
$={\left(\frac{3}{10}\right)}^{2}×\frac{7}{10}+\frac{7}{10}×{\left(\frac{3}{10}\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{126}{1000}$
Hence, required probability is $\frac{126}{1000}$.

#### Question 1:

A bag contains 6 black and 3 white balls. Another bag contains 5 black and 4 white balls. If one ball is drawn from each bag, find the probability that these two balls are of the same colour.

#### Question 2:

A bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black.

#### Question 3:

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both the balls are red. (ii) first ball is black and second is red. (iii) one of them is black and other is red.

#### Question 4:

Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of exactly one ace.

#### Question 5:

A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in narrating the same incident?

#### Question 6:

Kamal and Monica appeared for an interview for two vacancies. The probability of Kamal's selection is 1/3 and that of Monika's selection is 1/5. Find the probability that
(i) both of them will be selected
(ii) none of them will be selected
(iii) at least one of them will be selected
(iv) only one of them will be selected.

#### Question 7:

A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black?

#### Question 8:

A bag contains 8 red and 6 green balls. Three balls are drawn one after another without replacement. Find the probability that at least two balls drawn are green.

#### Question 9:

Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that to Tarun's rejection is 2/3. Find the probability that at least one of them will be selected.

#### Question 10:

A and B toss a coin alternately till one of them gets a head and wins the game. If A starts the game, find the probability that B will win the game.

#### Question 11:

Two cards are drawn from a well shuffled pack of 52 cards, one after another without replacement. Find the probability that one of these is red card and the other a black card?

#### Question 12:

Tickets are numbered from 1 to 10. Two tickets are drawn one after the other at random. Find the probability that the number on one of the tickets is a multiple of 5 and on the other a multiple of 4.

#### Question 13:

In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on the same fact.

It is given that the husband lies in 30% of the cases, while the wife lies in 35% cases

#### Question 14:

A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. What is the probability that
(i) both of them will be selected?
(ii) only one of them will be selected?
(iii) none of them will be selected?

#### Question 15:

A bag contains 7 white, 5 black and 4 red balls. Four balls are drawn without replacement. Find the probability that at least three balls are black.

#### Question 16:

A, B, and C are independent witness of an event which is known to have occurred. A speaks the truth three times out of four, B four times out of five and C five times out of six. What is the probability that the occurrence will be reported truthfully by majority of three witnesses?

#### Question 17:

A bag contains 4 white balls and 2 black balls. Another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that
(i) both are white
(ii) both are black
(iii) one is white and one is black

#### Question 18:

A bag contains 4 white, 7 black and 5 red balls. 4 balls are drawn with replacement. What is the probability that at least two are white?

#### Question 19:

Three cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that the cards are a king, a queen and a jack.

#### Question 20:

A bag contains 4 red and 5 black balls, a second bag contains 3 red and 7 black balls. One ball is drawn at random from each bag, find the probability that the (i) balls are of different colours (ii) balls are of the same colour.

#### Question 21:

A can hit a target 3 times in 6 shots, B : 2 times in 6 shots and C : 4 times in 4 shots. They fix a volley. What is the probability that at least 2 shots hit?

#### Question 22:

The probability of student A passing an examination is 2/9 and of student B passing is 5/9. Assuming the two events : 'A passes', 'B passes' as independent, find the probability of : (i) only A passing the examination (ii) only one of them passing the examination.

#### Question 23:

There are three urns A, B, and C. Urn A contains 4 red balls and 3 black balls. urn B contains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. One ball is drawn from each of these urns. What is the probability that 3 balls drawn consists of 2 red balls and a black ball?

#### Question 24:

X is taking up subjects - Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets
(i) Grade A in all subjects
(ii) Grade A in no subject
(iii) Grade A in two subjects.

#### Question 25:

A and B take turns in throwing two dice, the first to throw 9 being awarded the prize. Show that their chance of winning are in the ratio 9:8.

#### Question 26:

A, B and C in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely?

#### Question 27:

Three persons A, B, C throw a die in succession till one gets a 'six' and wins the game. Find their respective probabilities of winning.

#### Question 28:

A and B take turns in throwing two dice, the first to throw 10 being awarded the prize, show that if A has the first throw, their chance of winning are in the ratio 12 : 11.

#### Question 29:

There are 3 red and 5 black balls in bag 'A'; and 2 red and 3 black balls in bag 'B'. One ball is drawn from bag 'A' and two from bag 'B'. Find the probability that out of the 3 balls drawn one is red and 2 are black.

#### Question 30:

Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection is $\frac{1}{7}$ and that of John's selection is $\frac{1}{5}$. What is the probability that
(i) both of them will be selected?
(ii) only one of them will be selected?
(iii) none of them will be selected?

#### Question 31:

A bag contains 8 marbles of which 3 are blue and 5 are red. One marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be
(i) blue followed by red.
(ii) blue and red in any order.
(iii) of the same colour.

#### Question 32:

An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting
(i) 2 red balls
(ii) 2 blue balls
(iii) One red and one blue ball.

#### Question 34:

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among 100 students, what is the probability that: (i) you both enter the same section? (ii) you both enter the different sections?

#### Question 35:

In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the refree asked both the captains to throw a die alternately and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the refree was fair or not.

It can be seen that the probability that team A wins is not equal to the probability that team B wins.
Thus, the decision of the referee was not fair.

#### Question 36:

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.

Total of 7 on the dice can be obtained in the following ways:

(1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)

Probability of getting a total of 7 = $\frac{6}{36}=\frac{1}{6}$

Probability of not getting a total of 7 = $1-\frac{1}{6}=\frac{5}{6}$

Total of 10 on the dice can be obtained in the following ways:

(4, 6), (6, 4), (5, 5)

Probability of getting a total of 10 = $\frac{3}{36}=\frac{1}{12}$

Probability of not getting a total of 10 = $1-\frac{1}{12}=\frac{11}{12}$

Let E and F be the two events, defined as follows:

E = Getting a total of 7 in a single throw of a dice

F = Getting a total of 10 in a single throw of a dice

P(E) = $\frac{1}{6}$, $\mathrm{P}\left(\overline{\mathrm{E}}\right)=\frac{5}{6}$, P(F) = $\frac{1}{12}$, $\mathrm{P}\left(\overline{\mathrm{F}}\right)=\frac{11}{12}$

A wins if he gets a total of 7 in 1st, 3rd or 5th ... throws.

Probability of A getting a total of 7 in the 1st throw = $\frac{1}{6}$

A will get the 3rd throw if he fails in the 1st throw and B fails in the 2nd throw.

Probability of A getting a total of 7 in the 3rd throw =

Similarly, probability of getting a total of 7 in the 5th throw = and so on
Probability of winning of A = $\frac{1}{6}$+ $\left(\frac{5}{6}×\frac{11}{12}×\frac{1}{6}\right)$+ $\left(\frac{5}{6}×\frac{11}{12}×\frac{5}{6}×\frac{11}{12}×\frac{1}{6}\right)+...$  $=\frac{\frac{1}{6}}{1-\frac{5}{6}×\frac{11}{12}}=\frac{12}{17}\phantom{\rule{0ex}{0ex}}$
∴ Probability of winning of B = 1 − Probability of winning of A =  $1-\frac{12}{17}=\frac{5}{17}$

#### Question 37:

A and B throw a pair of dice alternatively till one of then gets the sum of the numbers as multiples of 6 and wins the game. If A starts first, find the probability of B winning the game.

Here,
Total outcomes = 36
Sum of numbers as multiple of 6 =
P (Getting sum multiple of 6) = $\frac{6}{36}$
= $\frac{1}{6}$
P (not-getting sum multiple of 6) = $1-\frac{1}{6}$
= $\frac{5}{6}$
P ( A winning game) = $\frac{1}{6}+\frac{5}{6}×\frac{5}{6}×\frac{1}{6}+\frac{5}{6}×\frac{5}{6}×\frac{5}{6}×\frac{5}{6}×\frac{1}{6}+.....................$
= $\frac{\frac{1}{6}}{1-\frac{25}{36}}=\frac{6}{11}$
$\therefore$P ( B winning game) = 1 $-$ P (A winning game)
$=1-\frac{6}{11}\phantom{\rule{0ex}{0ex}}=\frac{5}{11}$

#### Question 1:

A bag A contains 5 white and 6 black balls. Another bag B contains 4 white and 3 black balls. A ball is transferred from bag A to the bag B and then a ball is taken out of the second bag. Find the probability of this ball being black.

A black ball can be drawn in two mutually exclusive ways:

(I) By transferring a white ball from bag A to bag B, then drawing a black ball
(II) By transferring a black ball from bag A to bag B, then drawing a black ball

Let E1, E2 and A be the events as defined below:

E1 = A white ball is transferred from bag A to bag B
E2 = A black ball is transferred from bag A to bag B
A = A black ball is drawn

#### Question 2:

A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled at random from one of the two purses, what is the probability that it is a silver coin?

A silver coin can be drawn in two mutually exclusive ways:

(I) Selecting purse I and then drawing a silver coin from it
(II) Selecting purse II and then drawing a silver coin from it

Let E1, E2 and A be the events as defined below:

E1 = Selecting purse I
E2 = Selecting purse II
A = Drawing a silver coin

It is given that one of the purses is selected randomly.

#### Question 3:

One bag contains 4 yellow and 5 red balls. Another bag contains 6 yellow and 3 red balls. A ball is transferred from the first bag to the second bag and then a ball is drawn from the second bag. Find the probability that ball drawn is yellow.

A yellow ball can be drawn in two mutually exclusive ways:

(I) By transferring a red ball from first to second bag, then drawing a yellow ball
(II) By transferring a yellow ball from first to second bag, then drawing a yellow ball

Let E1, E2 and A be the events as defined below:

E1 = A red ball is transferred from first to second bag
E2 = A yellow ball is transferred from first to second bag
A =  A yellow ball is drawn

#### Question 4:

A bag contains 3 white and 2 black balls and another bag contains 2 white and 4 black balls. One bag is chosen at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is white.

A white ball can be drawn in two mutually exclusive ways:

(I) Selecting bag I and then drawing a white ball from it
(II) Selecting bag II and then drawing a white ball from it

Let E1, E2 and A be the events as defined below:
E1 = Selecting bag I
E2 = Selecting bag II
A = Drawing a white ball

It is given that one of the bags is selected randomly.

#### Question 5:

The contents of three bags I, II and III are as follows:
Bag I : 1 white, 2 black and 3 red balls,
Bag II : 2 white, 1 black and 1 red ball;
Bag III : 4 white, 5 black and 3 red balls.
A bag is chosen at random and two balls are drawn. What is the probability that the balls are white and red?

A white ball and a red ball can be drawn in three mutually exclusive ways:

(I) Selecting bag I and then drawing a white and a red ball from it
(II) Selecting bag II and then drawing a white and a red ball from it
(II) Selecting bag III and then drawing a white and a red ball from it

Let E1, E2 and A be the events as defined below:
E1 = Selecting bag I
E2 = Selecting bag II
E3 = Selecting bag II
A = Drawing a white and a red ball

It is given that one of the bags is selected randomly.

#### Question 6:

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the sum of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8?

Let E1, E2 and A be the events as defined below:
E1 = The coin shows a head
E2 = The coin shows a head
A = The noted number is 7 or 8

#### Question 7:

A factory has two machines A and B. Past records show that the machine A produced 60% of the items of output and machine B produced 40% of the items. Further 2% of the items produced by machine A were defective and 1% produced by machine B were defective. If an item is drawn at random, what is the probability that it is defective?

Let A, E1 and E2 denote the events that the item is defective, machine A is selected and machine B is selected, respectively.

#### Question 8:

The bag A contains 8 white and 7 black balls while the bag B contains 5 white and 4 black balls. One ball is randomly picked up from the bag A and mixed up with the balls in bag B. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

A white ball can be drawn in two mutually exclusive ways:

(I) By transferring a black ball from bag A to bag B, then drawing a white ball
(II) By transferring a white ball from bag A to bag B, then drawing a white ball

Let E1, E2 and A be events as defined below:
E1 = A black ball is transferred from bag A to bag B
E2 = A white ball is transferred from bag A to bag B
A = A white ball is drawn

#### Question 9:

A bag contains 4 white and 5 black balls and another bag contains 3 white and 4 black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is taken out from the latter. Find the probability that the ball drawn is white.

A white ball can be drawn in two mutually exclusive ways:

(I) By transferring a black ball from first to second bag, then drawing a white ball
(II) By transferring a white ball from first to second bag, then drawing a white ball

Let E1, E2 and A be the events as defined below:
E1 = A black ball is transferred from first to second bag
E2 = A white ball is transferred from first to second bag
A = A white ball is drawn

#### Question 10:

One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.

A white ball can be drawn in two mutually exclusive ways:

(I) By transferring a black ball from first to second bag, then drawing a white ball
(II) By transferring a white ball from first to second bag, then drawing a white ball

Let E1, E2 and A be the events as defined below:
E1 = A black ball is transferred from first to second bag
E2 = A white ball is transferred from first to second bag
A = A white ball is drawn

#### Question 11:

An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that its is a white ball.

A white ball can be drawn in three mutually exclusive ways:

(I) By transferring two black balls from first to second urn, then drawing a white ball
(II) By transferring two white balls from first to second urn, then drawing a white ball
(III) By transferring a white and a black ball from first to second urn, then drawing a white ball

Let E1, E2, E3 and A be the events as defined below:
E1 = Two black balls are transferred from first to second bag
E2 = Two white balls are transferred from first to second bag
E2 = A white and a black ball is transferred from first to second bag
A = A white ball is drawn

#### Question 12:

A bag contains 6 red and 8 black balls and another bag contains 8 red and 6 black balls. A ball is drawn from the first bag and without noticing its colour is put in the second bag. A ball is drawn from the second bag. Find the probability that the ball drawn is red in colour.

A red ball can be drawn in two mutually exclusive ways:

(I) By transferring a black ball from first to second bag, then drawing a red ball
(II) By transferring a red ball from first to second bag, then drawing a red ball

Let E1, E2 and A be the events as defined below:
E1 = A black ball is transferred from first to second bag
E2 = A red ball is transferred from first to second bag
A = A red ball is drawn

#### Question 13:

Three machines E1, E2, E3 in a certain factory produce 50%, 25% and 25%, respectively, of the total daily output of electric bulbs. It is known that 4% of the tubes produced one each of the machines Eand E2 are defective, and that 5% of those produced on E3 are defective. If one tube is picked up at random from a day's production, then calculate the probability that it is defective.
[NCERT EXEMPLAR, CBSE 2015]

Let A be the event that the tube picked is defective.

So, the probability that the picked tube is defective is $\frac{17}{400}$.

#### Question 1:

The contents of urns I, II, III are as follows:
Urn I : 1 white, 2 black and 3 red balls
Urn II : 2 white, 1 black and 1 red balls
Urn III : 4 white, 5 black and 3 red balls.
One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns I, II, III?

Let E1, E2 and E3 denote the events of selecting Urn I, Urn II and Urn III, respectively.

Let A be the event that the two balls drawn are white and red.

#### Question 2:

A bag A contains 2 white and 3 red balls and a bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag B.

Let A, E1 and E2 denote the events that the ball is red, bag A is chosen and bag B is chosen, respectively.

#### Question 3:

Three urns contains 2 white and 3 black balls; 3 white and 2 black balls and 4 white and 1 black ball respectively. One ball is drawn from an urn chosen at random and it was found to be white. Find the probability that it was drawn from the first urn.

Let E1, E2 and E3 denote the events of selecting Urn I, Urn II and Urn III, respectively.

Let A be the event that the ball drawn is white.

#### Question 4:

The contents of three urns are as follows:
Urn 1 : 7 white, 3 black balls, Urn 2 : 4 white, 6 black balls, and Urn 3 : 2 white, 8 black balls. One of these urns is chosen at random with probabilities 0.20, 0.60 and 0.20 respectively. From the chosen urn two balls are drawn at random without replacement. If both these balls are white, what is the probability that these came from urn 3?

Let E1, E2 and E3 denote the events of selecting Urn I, Urn II and Urn III, respectively.

Let A be the event that the two balls drawn are white.

#### Question 5:

Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a 'head' or 'tail' is obtained. If she obtained exactly one 'tail', then what is the probability that she threw 3, 4, 5 or 6 with the die?                                                                                                                                                                                    [CBSE 2015]

Let E1 be the event that the outcome on the die is 1 or 2 and E2 be the event that the outcome on the die is 3, 4, 5 or 6. Then,

Let A be the event of getting exactly one 'tail'.

P(A|E1) = Probability of getting exactly one tail by tossing the coin three times if she gets 1 or 2 = $\frac{3}{8}$

P(A|E2) = Probability of getting exactly one tail in a single throw of a coin if she gets 3, 4, 5 or 5 = $\frac{1}{2}$

As, the probability that the girl threw 3, 4, 5 or 6 with the die, if she obtained exactly one tail, is given by P(E2|A).

So, by using Baye's theorem, we get

$\mathrm{P}\left({E}_{2}|A\right)=\frac{\mathrm{P}\left({E}_{2}\right)×\mathrm{P}\left(A|{E}_{2}\right)}{\mathrm{P}\left({E}_{1}\right)×\mathrm{P}\left(A|{E}_{1}\right)+\mathrm{P}\left({E}_{2}\right)×\mathrm{P}\left(A|{E}_{2}\right)}\phantom{\rule{0ex}{0ex}}=\frac{\left(\frac{2}{3}×\frac{1}{2}\right)}{\left(\frac{1}{3}×\frac{3}{8}+\frac{2}{3}×\frac{1}{2}\right)}\phantom{\rule{0ex}{0ex}}=\frac{\left(\frac{2}{6}\right)}{\left(\frac{1}{8}+\frac{1}{3}\right)}\phantom{\rule{0ex}{0ex}}=\frac{\left(\frac{2}{6}\right)}{\left(\frac{11}{24}\right)}\phantom{\rule{0ex}{0ex}}=\frac{24×2}{11×6}\phantom{\rule{0ex}{0ex}}=\frac{8}{11}$

So, the probability that she threw 3, 4, 5 or 6 with the die if she obtained exactly one tail is $\frac{8}{11}$.

#### Question 6:

Two groups are competing for the positions of the Board of Directors of a Corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

Let E1 and E2 denote the events that the first group and the second group win the competition, respectively. Let A be the event of introducing a new product.

P(E1) = Probability that the first group wins the competition = 0.6

P(E2) = Probability that the second group wins the competition = 0.4

P(A/E1) = Probability of introducing a new product if the first group wins = 0.7

P(A/E2) = Probability of introducing a new product if the second group wins = 0.3

The probability that the new product is introduced by the second group is given by P(E2/A).

Using Bayes’ theorem, we get

#### Question 7:

Suppose 5 men out of 100 and 25 women out of 1000 are good orators. An orator is chosen at random. Find the probability that a male person is selected. Assume that there are equal number of men and women.

Let A, E1 and E2 denote the events that the person is a good orator, is a man and is a woman, respectively.

#### Question 8:

A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from
(i) LONDON (ii) CLIFTON?

Let A, E1 and E2 denote the events that the two consecutive letters are visible, the letter has come from LONDON and the letter has come from CLIFTON, respectively.

#### Question 9:

In a class, 5% of the boys and 10% of the girls have an IQ of more than 150. In this class, 60% of the students are boys. If a student is selected at random and found to have an IQ of more than 150, find the probability that the student is a boy.

Let A, E1 and E2 denote the events that the IQ is more than 150, the selected student is a boy and the selected student is a girl, respectively.

#### Question 10:

A factory has three machines X, Y and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?

Let E1, E2 and E3 denote the events that machine X produces bolts, machine Y produces bolts and machine Z produces bolts, respectively.

Let A be the event that the bolt is defective.

Total number of bolts = 1000 + 2000 + 3000 = 6000

P(E1) = $\frac{1000}{6000}=\frac{1}{6}$

P(E2) =$\frac{2000}{6000}=\frac{1}{3}$

P(E3) = $\frac{3000}{6000}=\frac{1}{2}$

The probability that the defective bolt is produced by machine X is given by P (E1/A).

#### Question 11:

An insurance company insured 3000 scooters, 4000 cars and 5000 trucks. The probabilities of the accident involving a scooter, a car and a truck are 0.02, 0.03 and 0.04 respectively. One of the insured vehicles meet with an accident. Find the probability that it is a (i) scooter (ii) car (iii) truck.

Let E1, E2 and E3 denote the events that the vehicle is a scooter, a car and a truck, respectively.

Let A be the event that the vehicle meets with an accident.

It is given that there are 3000 scooters, 4000 cars and 5000 trucks.

Total number of vehicles = 3000 + 4000 + 5000 = 12000

P(E1) = $\frac{3000}{12000}=\frac{1}{4}$

P(E2) =$\frac{4000}{12000}=\frac{1}{3}$

P(E3) = $\frac{5000}{12000}=\frac{5}{12}$

The probability that the vehicle, which meets with an accident, is a scooter is given by P (E1/A).

#### Question 12:

Suppose we have four boxes A, B, C, D containing coloured marbles as given below:
Figure

One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A? box B? box C?

Let R be the event of drawing the red marble.

Let EA, EB and EC denote the events of selecting box A, box B and box C, respectively.

Total number of marbles = 40

Number of red marbles = 15

Probability of drawing a red marble from box A is given by P(EA/R).

Probability of drawing a red marble from box B is given by P(EB/R).

Probability of drawing a red marble from box C is given by P(EC/R).

#### Question 13:

A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B on the job for 30% of the time and C on the job for 20% of the time. A defective item is produced. What is the probability that it was produced by A?

Let E1, E2 and E3 be the time taken by machine operators A, B, and C, respectively.

Let X be the event of producing defective items.

#### Question 14:

An item is manufactured by three machines A, B and C. Out of the total number of items manufactured during a specified period, 50% are manufactured on machine A, 30% on B and 20% on C. 2% of the items produced on A and 2% of items produced on B are defective and 3% of these produced on C are defective. All the items stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?                                                                                                  [NCERT EXEMPLAR]

So, the probability that the defective item drawn was manufactured on machine A is $\frac{5}{11}$.

#### Question 15:

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?                                                                                                               [CBSE 2014]

So, the probability that the head shown was of a two-headed coin is $\frac{20}{47}$.

Disclaimer: The answer given in the book is incorrect. The same has been corrected here.

#### Question 16:

In a factory, machine A produces 30% of the total output, machine B produces 25% and the machine C produces the remaining output. If defective items produced by machines A, B and C are 1%, 1.2%, 2% respectively. Three machines working together produce 10000 items in a day. An item is drawn at random from a day's output and found to be defective. Find the probability that it was produced by machine B?

Let A, E1, E2 and E3 denote the events that the item is defective, machine A is chosen, machine B is chosen and machine C is chosen, respectively.

#### Question 17:

A company has two plants to manufacture bicycles. The first plant manufactures 60% of the bicycles and the second plant 40%. Out of the 80% of the bicycles are rated of standard quality at the first plant and 90% of standard quality at the second plant. A bicycle is picked up at random and found to be standard quality. Find the probability that it comes from the second plant.

Let A, E1 and E2 denote the events that the cycle is of standard quality, plant I is chosen and plant II is chosen, respectively.

#### Question 18:

Three urns A, B and C contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.

Let A, E1 and E2 denote the events that the ball is red, bag A is chosen, bag B is chosen and bag C is chosen, respectively.

#### Question 19:

In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?

Let A, E1, E2 and E3 denote the events that the person suffers from the disease, is a smoker and a non-vegetarian, is a smoker and a vegetarian and the person is a non-smoker and a vegetarian, respectively.

#### Question 20:

A factory has three machines A, B and C, which produce 100, 200 and 300 items of a particular type daily. The machines produce 2%, 3% and 5% defective items respectively. One day when the production was over, an item was picked up randomly and it was found to  be defective. Find the probability that it was produced by machine A.

Let A, E1, E2 and E3 denote the events that the item is defective, machine A is chosen, machine B is chosen and machine C is chosen, respectively.

#### Question 21:

A bag contains 1 white and 6 red balls, and a second bag contains 4 white and 3 red balls. One of the bags is picked up at random and a ball is randomly drawn from it, and is found to be white in colour. Find the probability that the drawn ball was from the first bag.

Let A, E1 and E2 denote the events that the ball is white, bag I is chosen and bag II is chosen, respectively.

#### Question 22:

In a certain college, 4% of boys and 1% of girls are taller than 1.75 metres. Further more, 60% of the students in the colleges are girls. A student selected at random from the college is found to be taller than 1.75 metres. Find the probability that the selected students is girl.