Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable denote the number of rotten apples. If and represent mean and variance of , respectively, then is equal t

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Solution

Q.
Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable $X$ denote the number of rotten apples. If $μ$ and $σ^{2}$ represent mean and variance of $X$ , respectively, then $10 (μ^{2} + σ^{2})$ is equal to [2023]

1 250

2 20

3 25

4 30

Ans.
(2)

Let random variable $X$ denote the number of rotten apples.

$X_{i}$ $P_{i}$

$X = 0$ $\frac{{}^{7}{C_{4}} \times {}^{3}C_{0}}{{}^{10}{C_{4}}} = \frac{1}{6}$

$X = 1$ $\frac{{}^{3}{C_{1}} \times {}^{7}C_{3}}{{}^{10}{C_{4}}} = \frac{1}{2}$

$X = 2$ $\frac{{}^{3}{C_{2}} \times {}^{7}C_{2}}{{}^{10}{C_{4}}} = \frac{3}{10}$

$X = 3$ $\frac{{}^{3}{C_{3}} \times {}^{7}C_{1}}{{}^{10}{C_{4}}} = \frac{1}{30}$

$Mean (μ) = \sum X_{i} P_{i}$

$= 0 \times \frac{1}{6} + 1 \times \frac{1}{2} + 2 \times \frac{3}{10} + 3 \times \frac{1}{30} = \frac{6}{5}$

$Variance (σ^{2}) = \sum X_{i}^{2} P_{i} - μ^{2}$

$= 0 \times \frac{1}{6} + 1 \times \frac{1}{2} + 4 \times \frac{3}{10} + 9 \times \frac{1}{30} - \frac{36}{25} = \frac{14}{25}$

$So, 10 (μ^{2} + σ^{2}) = 10 (\frac{36}{25} + \frac{14}{25}) = 20$

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