Question

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 σ² represent mean and variance of X, respectively, then 10(μ²+σ²) is equal to

• A. 25
• B. 250
• C. 30
• D. 20

Answer 1

The Correct Option is D

 To solve this problem, we need to calculate the mean (μ) and variance (σ²) of the random variable X, which denotes the number of rotten apples drawn. Let's go through the solution step-by-step.

Firstly, determine the total number of apples:

• Total apples: 3 rotten + 7 good = 10 apples.

We are drawing 4 apples, and X is the number of rotten apples found in these 4 draws.

The expected value (mean, μ) of a hypergeometric distribution is given by:

\[\mu = \frac{n \cdot K}{N}\]

• \(n = 4\) (number of draws).
• \(K = 3\) (number of rotten apples).
• \(N = 10\) (total number of apples).

Substituting these values, we get:

\[\mu = \frac{4 \cdot 3}{10} = 1.2\]

Next, calculate the variance (σ²) of the hypergeometric distribution, which is given by:

\[\sigma^2 = \frac{n \cdot K \cdot (N-K) \cdot (N-n)}{N^2 \cdot (N-1)}\]

Substituting the known values:

\[\sigma^2 = \frac{4 \cdot 3 \cdot (10-3) \cdot (10-4)}{10^2 \cdot 9} = \frac{4 \cdot 3 \cdot 7 \cdot 6}{100 \cdot 9}\]

Calculating this, we get:

\[\sigma^2 = \frac{504}{900} = 0.56\]

Now, calculate \(10(\mu^2 + \sigma^2)\):

\[\mu^2 = 1.2^2 = 1.44\]

Substituting in the formula gives:

\[10(\mu^2 + \sigma^2) = 10(1.44 + 0.56) = 10 \times 2 = 20\]

Thus, \(10(\mu^2 + \sigma^2) = 20\), which corresponds to option

20

.

 

Therefore, the correct answer is 20.