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#### Question 1:

A coin is tossed 1000 times with the following frequencies:
Compute the probability for each event.

The coin is tossed 1000 times. So, the total number of trials is 1000.

Let A be the event of getting a head and B be the event of getting a tail.

The number of times A happens is 455 and the number of times B happens is 545.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

#### Question 2:

Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
One tail: 290 times
Find the probability of occurrence of each of these events.

The total number of trials is 500.

Let A be the event of getting two heads, B be the event of getting one tail and C be the event of getting no head.

The number of times A happens is 95, the number of times B happens is 290 and the number of times C happens is 115.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

#### Question 3:

Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:

If the three coins are simultaneously tossed again, compute the probability of:
(iii) at least one head coming up.
(iv) getting more heads than tails.
(v) getting more tails than heads.

The total number of trials is 100.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A be the event of getting two heads.

The number of times A happens is 36.

Therefore, we have

(ii) Let B be the event of getting three heads

The number of times B happens is 12.

Therefore, we have

(iii) Let C be the event of getting at least one head.

The number of times C happens is.

Therefore, we have

(iv) Let D be the event of getting more heads than tails.

The number of times D happens is.

Therefore, we have

(v) Let E be the event of getting more tails than heads.

The number of times E happens is.

Therefore, we have

#### Question 4:

1500 families with 2 children were selected randomly and the following data were recorded:

 Number of girls in a family 0 1 2 Number of families 211 814 475

If a family is chosen at random, compute the probability that it has:
(i) No girl
(ii) 1 girl
(iii) 2 girls
(iv) at most one girl
(v) more girls than boys

The total number of trials is 1500.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A be the event of having no girl.

The number of times A happens is 211.

Therefore, we have

(ii) Let B be the event of having one girl.

The number of times B happens is 814.

Therefore, we have

(iii) Let C be the event of having two girls.

The number of times C happens is 475.

Therefore, we have

(iv) Let D be the event of having at most one girl.

The number of times D happens is.

Therefore, we have

(v) Let E be the event of having more girls than boys.

The number of times E happens is 475.

Therefore, we have

#### Question 5:

In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays.
(i) he hits boundary
(ii) he does not hit a boundary.

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A be the event of hitting boundary.

The number of times A happens is 6.

Therefore, we have

(ii) Let B be the event of does not hitting boundary.

The number of times B happens is.

Therefore, we have

#### Question 6:

The percentage of marks obtained by a student in monthly unit tests are given below:

 Unit test: I II III IV V Percentage of marks obtained: 69 71 73 68 76

Find the probability that the student gets:
(i) more than 70% marks
(ii) less than 70% marks
(iii) a distinction

The total number of trials is 5.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A be the event of getting more than 70% marks.

The number of times A happens is 3.

Therefore, we have

(ii) Let B be the event of getting less than 70% marks.

The number of times B happens is 2.

Therefore, we have

(iii) Let C be the event of getting a distinction.

The number of times C happens is 1.

Therefore, we have

#### Question 7:

To know the opinion of the students about Mathematics, a survey of 200 students was conducted. The data is recorded in the following table:

 Opinion: Like Dislike Number of students: 135 65

Find the probability that a student chosen at random (i) likes Mathematics (ii) does not like it.

The total number of trials is 200.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A be the event of liking mathematics.

The number of times A happens is 135.

Therefore, we have

(ii) Let B be the event of disliking mathematics.

The number of times B happens is 65.

Therefore, we have

#### Question 8:

The blood groups of 30 students of class IX are recorded as follows:

 A B O O AB O A O B A O B A O O A AB O A A O O AB B A O B A B O

A student is selected at random from the class from blood donation, Fin the probability that the blood group of the student chosen is:
(i) A
(ii) B
(iii) AB
(iv) O

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A1 be the event that the blood group of a chosen student is A.

The number of times A1 happens is 9.

Therefore, we have

(iii) Let A2 be the event that the blood group of a chosen student is B.

The number of times A2 happens is 6.

Therefore, we have

(iii) Let A3 be the event that the blood group of a chosen student is AB.

The number of times A3 happens is 3.

Therefore, we have

(iv) Let A4 be the event that the blood group of a chosen student is O.

The number of times A4 happens is 12.

Therefore, we have

#### Question 9:

Eleven bags of wheat flour, each marked 5 Kg, actually contained the following weights of flour (in kg):
4.97, 5.05, 5.08, 5.03, 5.00, 5.06, 5.08, 4.98, 5.04, 5.07, 5.00

Find the probability that any of these bags chosen at random contains more than 5 kg of flour.

The total number of trials is 11.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Let A1 be the event that the actual weight of a chosen bag contain more than 5 Kg of flour.

The number of times A1 happens is 7.

Therefore, we have

#### Question 10:

Following table shows the birth month of 40 students of class IX.

 Jan. Feb March April May June July Aug. Sept. Oct. Nov. Dec. 3 4 2 2 5 1 2 5 3 4 4 4

Find the probability that a student was born in August.

The total number of trials is 40.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Let A1 be the event that the birth month of a chosen student is august.

The number of times A1 happens is 5.

Therefore, we have

#### Question 11:

Given below is the frequency distribution table regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days.

 Conc. of SO2 0.00-0.04 0.04-0.08 0.08-0.12 0.12-0.16 0.16-0.20 0.20-0.24 No. days: 4 8 9 2 4 3

Find the probability of concentration of sulphur dioxide in the interval
0.12-0.16 on any of these days.

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Let A1 be the event that the concentration of sulphur dioxide in a day is 0.12-0.16 parts per million.

The number of times A1 happens is 2.

Therefore, we have

#### Question 12:

A company selected 2400 families at random and survey them to determine a relationship between income level and the number of vehicles in a home. The information gathered is listed in the table below:

 Monthly income: (in Rs) Vehicles per family 0 1 2 Above 2 Less than 7000 7000-10000 10000-13000 13000-16000 16000 or more 10 0 1 2 1 160 305 535 469 579 25 27 29 29 82 0 2 1 25 88

If a family is chosen, find the probability that family is:

(i) earning Rs10000-13000 per month and owning exactly 2 vehicles.
(ii) earning Rs 16000 or more per month and owning exactly 1 vehicle.
(iii) earning less than Rs 7000 per month and does not own any vehicle.
(iv) earning Rs 13000-16000 per month and owning more than 2 vehicle.
(v) owning not more than 1 vehicle
(vi) owning at least one vehicle.

The total number of trials is 2400.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A1 be the event that a chosen family earns Rs 10000-13000 per month and owns exactly 2 vehicles.

The number of times A1 happens is 29.

Therefore, we have

(ii) Let A2 be the event that a chosen family earns Rs 16000 or more per month and owns exactly 1 vehicle.

The number of times A2 happens is 579.

Therefore, we have

(iii) Let A3 be the event that a chosen family earns less than Rs 7000 per month and does not owns any vehicles.

The number of times A3 happens is 10.

Therefore, we have

(iv) Let A4 be the event that a chosen family earns Rs 13000-16000 per month and owns more than 2 vehicles.

The number of times A4 happens is 25.

Therefore, we have

(v) Let A5 be the event that a chosen family owns not more than 1 vehicle (may be 0 or 1). In this case the number of vehicles is independent of the income of the family.

The number of times A5 happens is

.

Therefore, we have

(vi) Let A6 be the event that a chosen family owns atleast 1 vehicle (may be 1 or 2 or above 2). In this case the number of vehicles is independent of the income of the family.

The number of times A6 happens is

.

Therefore, we have

#### Question 13:

The following table gives the life time of 400 neon lamps:

 Life time (in hours) 300-400 400-500 500-600 600-700 700-800 800-900 900-1000 Number of lamps: 14 56 60 86 74 62 48

A bulb is selected of random, Find the probability that the  the life time of the selected bulb is:
(i) less than 400
(ii) between 300 to 800 hours
(iii) at least 700 hours.

The total number of trials is 400.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A1 be the event that the lifetime of a chosen bulb is less than 400 hours.

The number of times A1 happens is 14.

Therefore, we have

(ii) Let A2 be the event that the lifetime of a chosen bulb is in between 300 to 800 hours.

The number of times A2 happens is.

Therefore, we have

.

(iii) Let A3 be the event that the lifetime of a chosen bulb is atleast 700 hours.

The number of times A3 happens is.

Therefore, we have

#### Question 14:

Given below is the frequency distribution of wages (in Rs) of 30 workers in a certain factory:

 Wages (in Rs) 110-130 130-150 150-170 170-190 190-210 210-230 230-250 No. of workers 3 4 5 6 5 4 3

A worker is selected at random. Find the probability that his wages are:
(i) less than Rs 150
(ii) at least Rs 210
(iii) more than or equal to 150 but less than Rs 210.

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

(i) Let A1 be the event that the wages of a worker is less than Rs 150.

The number of times A1 happens is.

Therefore, we have.

(ii) Let A2 be the event that the wages of a worker is atleast Rs 210.

The number of times A2 happens is.

Therefore, we have.

(iii) Let A3 be the event that the wages of a worker is more than or equal to Rs 150 but less than Rs 210.

The number of times A3 happens is.

Therefore, we have

#### Question 1:

Define a trial.

What is the meaning of trial?

The word trial means a test of performance, qualities, or suitability.

Definition:

Any particular performance of a random experiment is called a trial. That is, when we perform an experiment it is called a trial of the experiment.

By experiment or trial, we mean a random experiment unless otherwise specified. Where you are required to differentiate between a trial and an experiment, consider the experiment to be a larger entity formed by the combination of a number of trials.

To illustrate the definition, let us take examples:

1. In the experiment of tossing 4 coins, we may consider tossing each coin as a trial and therefore say that there are 4 trials in the experiment.

2. In the experiment of rolling a dice 5 times, we may consider each rolls as a trial and therefore say that there are 5 trials in the experiment.

Note that rolling a dice 5 times is same as rolling 5 dices each one time. Similarly, tossing 4 coins is same as tossing one coin 4 times.

#### Question 2:

Define an elementary event.

What are the meanings of elementary event?

The word elementary means simple, non decomposable into elements or other primary constituents and the word event means something that result.

Definition:

An elementary event is any single outcome of a trial. Elementary events are also called simple events.

To illustrate the definition, let us take examples:

1. In the experiment of tossing a coin, the possible outcomes H and T. Any one outcome like H is called an elementary event.

2. In the experiment of rolling a dice, the possible outcomes are 1, 2, 3, 4, 5 and 6. Any one outcome like 4 is called an elementary event.

Note that H stands for getting a head and T stands for getting a tail in the experiment of tossing a coin.

#### Question 3:

Define an event.

What are the meanings of event?

The word event means something that result.

Definition:

An event is a collection of outcomes of a trial of a random experiment.

To illustrate the definition, let us take examples:

1. When two coins are tossed simultaneously, the possible outcomes are HH, HT, TH and TT. Any one outcome like HH is called an event (elementary event). The collections like {HH, HT}, {HH, HT, TT} etc are all events (compound event).

2. In the experiment of rolling a dice, the possible outcomes are 1, 2, 3, 4, 5 and 6. Any one outcome like 4 is called an event (elementary event). The collections like {1, 2}, {1, 2, 3}, {2, 5, 6}, {2, 3, 4, 5} etc are all events (compound events).

Note that H stands for getting a head and T stands for getting a tail in the experiment of tossing a coin.

#### Question 4:

Define probability of an event.

The probability of an event denotes the relative frequency of occurrence of an experiment’s outcome, when repeating the experiment.

Definition:

The empirical or experimental definition of probability is that if n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials, then the probability of happening of event A is denoted byand is given by

To illustrate the definition, let us take examples:

1. When two coins are tossed simultaneously, the possible outcomes are HH, HT, TH and TT. The total number of trials is 4. Let A be the event of occurring exactly two heads. The number of times A happens is 1. So, the probability of the event A is

2. In the experiment of rolling a dice, the possible outcomes are 1, 2, 3, 4, 5 and 6. Let A be the event of occurring a number greater than 3. The total number of trials is 6. The number of times A happens is 3. So, the probability of the event A is

Note that H stands for getting a head and T stands for getting a tail in the experiment of tossing a coin.

#### Question 5:

A big contains 4 white balls and some red balls. If the probability of drawing a white ball from the bag is $\frac{2}{5}$, find the number of red balls in the bag.

The number of white balls is 4. Let the number of red balls is x. Then the total number of trials is.

Let A be the event of drawing a white ball.

The number of times A happens is 4.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have.

But, it is given that. So, we have

Hence the number of red balls is.

#### Question 6:

A die is thrown 100 times. If the probability of getting an even number is $\frac{2}{5}$. How many times an odd number is obtained?

The total number of trials is 100. Let the number of times an even number is obtained is x.

Let A be the event of getting an even number.

The number of times A happens is x.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have.

But, it is given that. So, we have

Hence an even number is obtained 40 times. Consequently, an odd number is obtainedtimes.

#### Question 7:

Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes:

Find the probability of getting at most two heads.

The total number of trials is 200.

Let A be the event of getting at most two heads.

The number of times A happens is.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have.

#### Question 8:

In Q.No. 7, what is the probability of getting at least two heads?

The total number of trials is 200.

Let A be the event of getting atleast two heads.

The number of times A happens is.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

#### Question 1:

Mark the correct alternative in each of the following:
The probability of an impossible event is

(a) 1
(b) 0
(c) less than 0
(d) greater than 1

We have to find the probability of an impossible event.

Note that the number of occurrence of an impossible event is 0. This is the reason that’s why it is called impossible event.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Note that n is a positive integer, it can’t be zero. So, whatever may be the value of n, the probability of an impossible event is.

Hence the correct option is (b).

#### Question 2:

The probability of a certain event is

(a) 0
(b) 1
(c) greater than 1
(d) less than 0

We have to find the probability of a certain event.

Note that the number of occurrence of an impossible event is same as the total number of trials. When we repeat the experiment, every times it occurs. This is the reason that’s why it is called certain event.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Note that n is a positive integer, it can’t be zero. So, the probability of an impossible event is.

Hence the correct option is (b).

#### Question 3:

The probability an event of a trial is

(a) 1
(b) 0
(c) less than 1
(d) more than 1

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Note that m is always less than or equal to n and n is a positive integers, it can’t be zero. But, m is a non negative integer. So, the maximum value of probability of an event is, which is the probability of a certain event and the minimum value of it is 0, which is the probability of an impossible event. For any other events the value is in between 0 and 1.

Hence the correct option is (c).

#### Question 4:

Which of the following cannot be the probability of an event?

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

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

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

(d) 1

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Note that m is always less than or equal to n and n is a positive integers, it can’t be zero. But, m is a non negative integer. So, the maximum value of probability of an event is, which is the probability of a certain event and the minimum value of it is 0, which is the probability of an impossible event. For any other events the value is in between 0 and 1.

All the options except (c) satisfy the above criteria’s.

Hence the correct option is (c).

#### Question 5:

Two coins are tossed simultaneously. The probability of getting atmost one head is

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

(b) $\frac{3}{4}$

(c) $\frac{1}{2}$

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

The random experiment is tossing two coins simultaneously.

All the possible outcomes are HH, HT, TH, and TT.

Let A be the event of getting at most one head.

The number of times A happens is 3.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

So, the correct choice is (b).

#### Question 6:

A coin is tossed 1000 times, if the probability of getting a tail is 3/8, how many times head is obtained?
(a) 525

(b) 375

(c) 625

(d) 725

The total number of trials is 1000. Let x be the number of times a tail occurs.

Let A be the event of getting a tail.

The number of times A happens is x.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have.

But, it is given that. So, we have

Hence a tail is obtained 375 times.

So, the correct choice is (c).

#### Question 7:

A dice is rolled 600 times and the occurrence of the outcomes 1, 2, 3, 4, 5 and 6 are given below:

 Outcome 1 2 3 4 5 6 Frequency 200 30 120 100 50 100

The probability of getting a prime number is

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

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

(c) $\frac{49}{60}$

(d) $\frac{39}{125}$

The total number of trials is 600.

Let A be the event of getting a prime number (2, 3 and5).

The number of times A happens is.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

So, the correct choice is (a).

#### Question 8:

The percentage of attendance of different classes in a year in a school is given below:

 Class: X IX VIII VII VI V Attendance: 30 62 85 92 76 55

What is the probability that the class attendance is more than 75%?
(a) $\frac{1}{6}$

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

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

(d) $\frac{1}{2}$

The total number of trials is 6.

Let A be the event that the attendance of a class is more than 75%.

The number of times A happens is 3 (for classes’ VIII, VII and VI).

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

So, the correct choice is (d).

#### Question 9:

A bag contains 50 coins and each coin is marked from 51 to 100. One coin is picked at random. The probability that the number on the coin is not a prime number, is

(a) $\frac{1}{5}$

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

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

(d) $\frac{4}{5}$

The total number of trials is 50.

Let A be the event that the number on the picked coin is not a prime.

The prime’s lies in between 51 and 100 are 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. They are 10 in numbers. Therefore the numbers lies between 51 and 100 and which are not primes are in numbers.

So, the number of times A happens is 40.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

So, the correct choice is (d).

#### Question 10:

In a football match, Ronaldo makes 4 goals from 10 penalty kicks. The probability of converting a penalty kick into a goal by Ronaldo, is

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

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

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

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

The total number of trials is 10.

Let A be the event that Ronaldo makes a goal in a penalty kick.

The number of times A happens is 4.

Remember the empirical or experimental or observed frequency approach to probability.

If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials. Then the empirical probability of happening of event A is denoted byand is given by

Therefore, we have

So, the correct choice is (d).

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