Two cards are drawn successively with replacement from a well shuffled deck of 52 cards, then the meanof the number of aces is

Options

A. `1//13`
B. `3//13`
C. `2//13`
D. none of these

Explanation

To find the mean of the number of aces when two cards are drawn successively with replacement from a well-shuffled deck of 52 cards, we can follow these steps:

### Step 1: Define the Random Variable
Let \( X \) be the random variable representing the number of aces drawn in the two draws.

### Step 2: Identify the Distribution
Since we are drawing cards with replacement, the number of aces follows a binomial distribution. This is because each draw is independent, and there are only two possible outcomes for each draw: either we draw an ace or we do not draw an ace.

### Step 3: Determine Parameters of the Binomial Distribution
In a binomial distribution, we have:
- \( n \): the number of trials (draws), which is 2 in this case.
- \( p \): the probability of success (drawing an ace) in each trial.

### Step 4: Calculate the Probability of Drawing an Ace
In a standard deck of 52 cards, there are 4 aces. Therefore, the probability \( p \) of drawing an ace is:
\[ p = \frac{\text{Number of Aces}}{\text{Total Number of Cards}} = \frac{4}{52} = \frac{1}{13} \]

### Step 5: Calculate the Mean of the Binomial Distribution
The mean \( \mu \) of a binomial distribution is given by the formula:
\[ \mu = n \times p \]
Substituting the values we have:
\[ \mu = 2 \times \frac{1}{13} = \frac{2}{13} \]

### Step 6: Conclusion
Therefore, the mean of the number of aces drawn when two cards are drawn successively with replacement is:
\[ \frac{2}{13} \]