From a pack of 52 cards, two cards are drawn, the first being replaced before the second is drawn. What is the probability that the first is a diamond and the second is a king ?

Options

A. `1/4`
B. `4/13`
C. `1/52`
D. `4/15`

Explanation

To solve the problem, we need to find the probability that the first card drawn is a diamond and the second card drawn is a king, given that the first card is replaced before the second card is drawn.

### Step 1: Determine the probability of drawing a diamond first.

- **Total number of cards in a deck**: 52
- **Number of diamond cards in a deck**: 13

The probability of drawing a diamond (P(D)) is given by the formula:

\[ P(D) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{13}{52} \]

Simplifying this fraction:

\[ P(D) = \frac{1}{4} \]

### Step 2: Determine the probability of drawing a king second.

Since the first card is replaced, the total number of cards remains 52 for the second draw.

- **Number of king cards in a deck**: 4

The probability of drawing a king (P(K)) is given by:

\[ P(K) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{52} \]

Simplifying this fraction:

\[ P(K) = \frac{1}{13} \]

### Step 3: Calculate the combined probability.

Since the two events (drawing a diamond first and drawing a king second) are independent (due to replacement), we can multiply the probabilities:

\[ P(D \text{ and } K) = P(D) \times P(K) = \frac{1}{4} \times \frac{1}{13} \]

Calculating this gives:

\[ P(D \text{ and } K) = \frac{1}{52} \]

### Final Answer:

The probability that the first card is a diamond and the second card is a king is \(\frac{1}{52}\).

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