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

We are given that there are 3 rotten apples and 7 good apples. 4 apples are drawn randomly without replacement. We need to find the value of \( 10(\mu^2 + \sigma^2) \), where \( \mu \) is the mean and \( \sigma^2 \) is the variance of the random variable \( X \), which represents the number of rotten apples drawn.

 Step 1: Defining the Problem.
The number of rotten apples drawn, \( X \), follows a hypergeometric distribution, since we are drawing without replacement. The probability mass function (PMF) of \( X \) for a hypergeometric distribution is given by: \[ P(X = k) = \frac{\binom{3}{k} \binom{7}{4-k}}{\binom{10}{4}}, \] where: - \( 3 \) is the number of rotten apples,
- \( 7 \) is the number of good apples,
- \( 4 \) is the total number of apples drawn,
- \( k \) is the number of rotten apples drawn.

Step 2: Mean and Variance of Hypergeometric Distribution.
For a hypergeometric distribution, the mean \( \mu \) and variance \( \sigma^2 \) are given by the formulas: \[ \mu = \frac{nK}{N}, \quad \sigma^2 = \frac{nK(N-K)(N-n)}{N^2(N-1)}, \] where: - \( n = 4 \) (number of draws),
- \( K = 3 \) (total number of rotten apples),
- \( N = 10 \) (total number of apples).
Calculating the Mean \( \mu \): \[ \mu = \frac{4 \times 3}{10} = \frac{12}{10} = 1.2. \] 
Calculating the Variance \( \sigma^2 \): \[ \sigma^2 = \frac{4 \times 3 \times (10 - 3) \times (10 - 4)}{10^2 \times (10 - 1)} = \frac{4 \times 3 \times 7 \times 6}{100 \times 9} = \frac{504}{900} = 0.56. \]

Step 3: Calculating \( 10(\mu^2 + \sigma^2) \). Now, calculate \( 10(\mu^2 + \sigma^2) \): \[ \mu^2 = (1.2)^2 = 1.44, \quad \sigma^2 = 0.56, \] \[ \mu^2 + \sigma^2 = 1.44 + 0.56 = 2. \] \[ 10(\mu^2 + \sigma^2) = 10 \times 2 = 20. \] 

Thus, the value of \( 10(\mu^2 + \sigma^2) \) is \( 20 \), and the correct answer is option (4).