Question

Sasha is working on a magic trick with a standard deck of cards. She chooses an ace, puts it back, then chooses a red card. What is P(ace|red)
? Round to the nearest tenth.(1 point)
percent

Answer 1

To find \( P(\text{ace} | \text{red}) \), we can use Bayes’ Theorem or consider the problem directly with conditional probability.

1. **Identify the total number of cards**:
A standard deck has 52 cards.

2. **Identify the relevant subsets**:
- There are 4 aces in a deck.
- There are 26 red cards (13 hearts and 13 diamonds).

3. **Calculate \( P(\text{red}) \)**:
The probability of selecting a red card is:
\[
P(\text{red}) = \frac{26}{52} = \frac{1}{2}
\]

4. **Calculate \( P(\text{ace} \cap \text{red}) \)**:
The probability of selecting an ace that is also red (the red aces) is:
\[
P(\text{ace} \cap \text{red}) = \frac{2}{52} = \frac{1}{26}
\]

5. **Now, use the formula for conditional probability**:
\[
P(\text{ace} | \text{red}) = \frac{P(\text{ace} \cap \text{red})}{P(\text{red})}
\]
Substituting the values we have:
\[
P(\text{ace} | \text{red}) = \frac{\frac{1}{26}}{\frac{1}{2}} = \frac{1}{26} \times \frac{2}{1} = \frac{2}{26} = \frac{1}{13}
\]

6. **Convert the probability to a percentage**:
\[
\frac{1}{13} \approx 0.07692
\]
To convert to a percentage, multiply by 100:
\[
0.07692 \times 100 \approx 7.692 \%
\]

7. **Round to the nearest tenth**:
Rounding 7.692% to the nearest tenth gives us:
\[
7.7\%
\]

Therefore, \( P(\text{ace} | \text{red}) \approx 7.7\% \).