Three rotten apples are mixed accidentally with seven good apples and four apples are draw…

Question

Three rotten apples are mixed accidentally 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  and $\sigma^2$ represent mean and variance of X, respectively, then $10(\mu^2+\sigma^2)$ is equal to

Accepted Answer

$X$ follows a hypergeometric distribution (3 rotten, 7 good, draw 4 of 10 without replacement).

$$\mu = \sum xP(x) = \frac{6}{2}\cdot\frac{1}{5} \quad(\text{i.e. } E[X] = \frac{4\times3}{10} = 1.2)$$

Using the given intermediate computation, $\sigma^2+\mu^2 = 0+\dfrac12+\dfrac{12}{10}+\dfrac{9}{30} = 2$.

$$10(\sigma^2+\mu^2) = 10\times2 = 20$$

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