To find \( P(\text{ace} | \text{red}) \), we can use Bayes' Theorem, which can be formulated as follows:
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]
Where:
- \( A \) is the event of drawing an ace.
- \( B \) is the event of drawing a red card.
Step 1: Calculate \( P(A) \)
There are 4 aces in a standard deck of 52 cards. Therefore:
\[ P(A) = \frac{4}{52} = \frac{1}{13} \]
Step 2: Calculate \( P(B) \)
There are 26 red cards in a standard deck (13 hearts and 13 diamonds). Therefore:
\[ P(B) = \frac{26}{52} = \frac{1}{2} \]
Step 3: Calculate \( P(B|A) \)
If we know that an ace has been drawn, the only red aces are the Ace of Hearts and the Ace of Diamonds, which gives us 2 favorable outcomes out of 4 possible aces. Therefore:
\[ P(B|A) = \frac{2}{4} = \frac{1}{2} \]
Step 4: Use Bayes' Theorem
Now we can substitute these values into Bayes’ theorem:
\[ P(\text{ace} | \text{red}) = \frac{P(\text{red} | \text{ace}) \cdot P(\text{ace})}{P(\text{red})} \] \[ P(\text{ace} | \text{red}) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{\left(\frac{1}{2}\right) \cdot \left(\frac{1}{13}\right)}{\frac{1}{2}} \]
Since \(P(B)\) cancels out:
\[ P(\text{ace} | \text{red}) = \frac{1}{13} \]
Step 5: Convert to decimal and round
Converting \( \frac{1}{13} \) to a decimal gives approximately 0.0769. Rounding to the nearest tenth gives:
\[ \approx 0.1 \]
Thus, \( P(\text{ace} | \text{red}) \) is approximately \( \boxed{0.1} \).