Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. Let X danote the random variable of number of aces obtained in the two drawn cards. Then, P(X = 1) + P(X = 2) equals

Options

A. `(25)/(169)`
B. `(52)/(169)`
C. `(49)/(169)`
D. `(24)/(169)`

Explanation

To solve the problem, we need to find \( P(X = 1) + P(X = 2) \) where \( X \) is the random variable representing the number of aces drawn in two successive draws from a deck of cards with replacement.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a standard deck of 52 cards, which contains 4 aces.
- We are drawing 2 cards with replacement, meaning after drawing a card, we put it back into the deck before drawing again.

2. **Defining Probabilities**:
- The probability of drawing an ace (success) in one draw is:
\[ P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} \]
- The probability of not drawing an ace (failure) is:
\[ P(\text{Not Ace}) = 1 - P(\text{Ace}) = 1 - \frac{1}{13} = \frac{12}{13} \]

3. **Calculating \( P(X = 1) \)**:
- \( P(X = 1) \) occurs when we draw exactly 1 ace in 2 draws. This can happen in two ways:
- Ace on the first draw and not an ace on the second draw.
- Not an ace on the first draw and an ace on the second draw.
- The probability for each scenario is:
- Ace first, not ace second: \( P(\text{Ace}) \cdot P(\text{Not Ace}) = \frac{1}{13} \cdot \frac{12}{13} \)
- Not ace first, ace second: \( P(\text{Not Ace}) \cdot P(\text{Ace}) = \frac{12}{13} \cdot \frac{1}{13} \)
- Therefore,
\[ P(X = 1) = 2 \cdot \left(\frac{1}{13} \cdot \frac{12}{13}\right) = 2 \cdot \frac{12}{169} = \frac{24}{169} \]

4. **Calculating \( P(X = 2) \)**:
- \( P(X = 2) \) occurs when we draw 2 aces in 2 draws. The probability is:
\[ P(X = 2) = P(\text{Ace}) \cdot P(\text{Ace}) = \frac{1}{13} \cdot \frac{1}{13} = \frac{1}{169} \]

5. **Adding Probabilities**:
- Now we can find \( P(X = 1) + P(X = 2) \):
\[ P(X = 1) + P(X = 2) = \frac{24}{169} + \frac{1}{169} = \frac{25}{169} \]

### Final Answer:
Thus, \( P(X = 1) + P(X = 2) = \frac{25}{169} \).