Question

Two cards are drawn at random one after the other with replacement from a pack of playing cards. If \( X \) is the random variable denoting the number of ace cards drawn, then the mean of the probability distribution of \( X \) is:

• A. \( 2 \)
• B. \( \frac{2}{13} \)
• C. \( 1 \)
• D. \( \frac{1}{13} \)

Hint:

For a binomial distribution \( X \sim \text{Bin}(n, p) \), the expected value is given by \( E(X) = np \). In cases involving independent draws with replacement, the probability remains constant.

Answer 1

The Correct Option is B

Step 1: Define the Probability Distribution 
A standard deck of playing cards consists of 52 cards, out of which 4 are aces. The probability of drawing an ace in a single draw is: \[ P(A) = \frac{4}{52} = \frac{1}{13} \] Since the draws are done with replacement, the probability of drawing an ace remains constant for both draws. 

Step 2: Define the Random Variable \( X \) 
The random variable \( X \) represents the number of aces drawn in two independent trials. Since each draw is independent and follows a Bernoulli process, \( X \) follows a binomial distribution: \[ X \sim \text{Binomial}(n=2, p=\frac{1}{13}) \] where: - \( n = 2 \) (two trials), - \( p = \frac{1}{13} \) (probability of success in a single trial). 

Step 3: Compute the Expected Value (Mean) 
The mean of a binomial distribution is given by: \[ E(X) = n p \] Substituting values: \[ E(X) = 2 \times \frac{1}{13} = \frac{2}{13} \] 

Step 4: Conclusion 
Thus, the mean of the probability distribution of \( X \) is: \[ \mathbf{\frac{2}{13}} \]