3 rotten apples are are mixed with 7 fresh appl, find the probability distribution of number of rotten apples 3 - Maths - Probability

Question

3 rotten apples are are mixed with 7 fresh appl, find the
probability distribution of number of rotten apples 3 apples are
drawn at random (without replacement) also find the mean of
distribution

Ajanta Trivedi · Accepted Answer

rotten apples =3 , fresh apples = 7, total number of apples= 10

since 3 apples are drawn at random without replacement from the total of 10 apples

let X denote the random variable number of rotten apples (out of the three apples drawn).

then X can take the values 0,1,2 or 3.

P(X=0)=P(getting no rotten apple)= $\frac{{}^{7}{C_{3}}}{{}^{10}{C_{3}}} = \frac{7 * 6 * 5}{3!} * \frac{3!}{10 * 9 * 8} = \frac{7}{24}$

P(X=1)=P(getting 1 rotten apple and 2 fresh apples)= $\frac{{}^{3}{C_{1}} * {}^{7}{C_{2}}}{{}^{10}{C_{3}}} = \frac{3 * \frac{7 * 6}{2}}{\frac{10 * 9 * 8}{3 * 2}} = \frac{3 * 3 * 7}{5 * 3 * 8} = \frac{21}{40}$

P(X=2)=P(getting 2 rotten apple and 1 fresh apple)= $\frac{{}^{3}{C_{2}} * {}^{7}{C_{1}}}{{}^{10}{C_{3}}} = \frac{3 * 7}{\frac{10 * 9 * 8}{3 * 2}} = \frac{3 * 7}{5 * 3 * 8} = \frac{7}{40}$

P(X=3)=P(getting all the 3 rotten apples i.e. no fresh apple)= $\frac{{}^{3}{C_{3}}}{{}^{10}{C_{3}}} = \frac{1}{120}$

∴ the probability distribution of the discrete random variable X is

X	0	1	2	3
P(X)	$\frac{7}{24}$	$\frac{21}{40}$	$\frac{7}{40}$	$\frac{1}{120}$

now the mean of the distribution $μ = \sum_{X = 0}^{3} X \cdot P (X)$

$μ = 0 \cdot \frac{7}{24} + 1 \cdot \frac{21}{40} + 2 \cdot \frac{7}{40} + 3 \cdot \frac{1}{120} = 0 + \frac{21}{40} + \frac{14}{40} + \frac{1}{40} = \frac{36}{40} = \frac{9}{10}$

therefore mean =9/10

3 rotten apples are are mixed with 7 fresh appl, find the probability distribution of number of rotten apples 3 apples are drawn at random (without replacement) . also find the mean of distribution....