From a well shuffled pack of playing cards, two cards are drawn one by one with replacement. The probability that both are aces is

Options

A. `(2)/(13)`
B. `(1)/(51)`
C. `(1)/(221)`
D. none

Answer

To find the probability that both cards drawn from a well-shuffled pack of playing cards are aces, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Number of Cards**:
A standard deck of playing cards contains 52 cards.

2. **Identify the Total Outcomes**:
When drawing one card from the deck, the total number of outcomes is 52 (since there are 52 cards).

3. **Determine the Favorable Outcomes for the First Draw**:
There are 4 aces in a deck of cards (Ace of Hearts, Ace of Diamonds, Ace of Clubs, Ace of Spades). Therefore, the number of favorable outcomes for drawing an ace is 4.

4. **Calculate the Probability of Drawing an Ace on the First Draw**:
The probability of drawing an ace on the first draw is given by the formula:
\[ P(\text{Ace on 1st draw}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{52} = \frac{1}{13} \]

5. **Replace the Card**:
Since the card is drawn with replacement, the total number of cards remains 52 for the second draw.

6. **Determine the Favorable Outcomes for the Second Draw**:
Again, there are still 4 aces in the deck.

7. **Calculate the Probability of Drawing an Ace on the Second Draw**:
The probability of drawing an ace on the second draw is the same as for the first draw:
\[ P(\text{Ace on 2nd draw}) = \frac{4}{52} = \frac{1}{13} \]

8. **Calculate the Combined Probability**:
Since the draws are independent (due to replacement), the combined probability of both events occurring (drawing an ace on the first draw and drawing an ace on the second draw) is:
\[ P(\text{Both Aces}) = P(\text{Ace on 1st draw}) \times P(\text{Ace on 2nd draw}) = \frac{1}{13} \times \frac{1}{13} = \frac{1}{169} \]

### Final Answer:
The probability that both cards drawn are aces is \(\frac{1}{169}\).

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