A man draws a card from a pack of 52 cards and then replaces it. After shuffling the pack, he again draws a card. This he repeats a number of times. The probability that he will draw a heart for the first time in the third draw is

Options

A. `(9)/(64)`
B. `(27)/(64)`
C. `(1)/(4)xx(.^(39)C_(2))/(.^(52)C_(2))`
D. none of these

Explanation

To solve the problem step by step, we will calculate the probability that a man draws a heart for the first time in the third draw from a pack of 52 cards.

### Step 1: Understand the Problem
We need to find the probability that the first heart is drawn on the third draw. This means that the first two draws must not be hearts, and the third draw must be a heart.

### Step 2: Define the Events
Let \( A_i \) be the event of drawing a heart on the \( i \)-th draw. The complement event \( A_i' \) (not drawing a heart) occurs when we draw any of the other 39 cards (since there are 52 cards in total and 13 of them are hearts).

### Step 3: Calculate the Probabilities
- The probability of drawing a heart on any draw is:
\[ P(A) = \frac{13}{52} = \frac{1}{4} \]
- The probability of not drawing a heart (drawing any other card) is:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Calculate the Required Probability
We want the probability of the event where:
- The first draw is not a heart: \( P(A_1') = \frac{3}{4} \)
- The second draw is not a heart: \( P(A_2') = \frac{3}{4} \)
- The third draw is a heart: \( P(A_3) = \frac{1}{4} \)

Since the draws are independent, we can multiply the probabilities:
\[ P(\text{first heart on third draw}) = P(A_1') \times P(A_2') \times P(A_3) = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{1}{4}\right) \]

### Step 5: Perform the Calculation
Calculating the above expression:
\[ P(\text{first heart on third draw}) = \frac{3}{4} \times \frac{3}{4} \times \frac{1}{4} = \frac{9}{64} \]

### Final Result
Thus, the probability that he will draw a heart for the first time in the third draw is:
\[ \frac{9}{64} \]