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) perc

Answer 1

To find \( P(\text{ace} \mid \text{red}) \), we can use the definition of conditional probability:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

Where:
- \( A \) is the event of choosing an ace.
- \( B \) is the event of choosing a red card.

### Step 1: Find \( P(\text{red}) \)

A standard deck of cards has 52 cards in total, 26 of which are red (13 hearts and 13 diamonds).

So,

\[
P(\text{red}) = \frac{26}{52} = \frac{1}{2}
\]

### Step 2: Find \( P(\text{ace} \cap \text{red}) \)

There are 2 red aces in a deck (the Ace of Hearts and the Ace of Diamonds).

So,

\[
P(\text{ace} \cap \text{red}) = \frac{2}{52} = \frac{1}{26}
\]

### Step 3: Use the conditional probability formula

Now we can substitute our values into the conditional probability formula:

\[
P(\text{ace} \mid \text{red}) = \frac{P(\text{ace} \cap \text{red})}{P(\text{red})} = \frac{\frac{1}{26}}{\frac{1}{2}} = \frac{1}{26} \times \frac{2}{1} = \frac{2}{26} = \frac{1}{13}
\]

### Step 4: Convert to decimal and round to the nearest tenth

Now we convert \( \frac{1}{13} \) to a decimal:

\[
\frac{1}{13} \approx 0.0769
\]

Rounded to the nearest tenth, we get:

\[
P(\text{ace} \mid \text{red}) \approx 0.1
\]

Thus, the final answer is:

\[
\boxed{0.1}
\]