JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Probability

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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

(A)20✓ Correct
(B)250
(C)25
(D)30

Answer: (A)

Solution

Drawing

4

apples from

3

rotten and

7

good, the count of rotten apples

X

is hypergeometric with

P (X = x) = \frac{( x 3 ) ( 4 - x 7 )}{( 4 10 )}

, giving

P (0) = \frac{1}{6}, P (1) = \frac{1}{2}, P (2) = \frac{3}{10}, P (3) = \frac{1}{30}

. The mean is

μ = \sum x P (x) = \frac{6}{5}

, and

E (X^{2}) = \sum x^{2} P (x) = 2

, so the variance is

σ^{2} = E (X^{2}) - μ^{2} = 2 - \frac{36}{25} = \frac{14}{25}

. Then

μ^{2} + σ^{2} = \frac{36}{25} + \frac{14}{25} = 2

, so

10 (μ^{2} + σ^{2}) = 20

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Single correct option

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JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Probability

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