Question

Margarita draws two cards from a standard deck of cards. Event A
is to draw any card from the suit of hearts without replacing it and event B
is to draw any red card. What is P(A∩B)
? Note that there are 26 red cards in a standard deck and that a suit includes 13 cards. Round your answer to the nearest tenths place.(1 point)

Answer 1

To find the probability of the event \( P(A \cap B) \), where event \( A \) is drawing a heart and event \( B \) is drawing a red card from a standard deck of 52 cards, we first identify the overlap between these events.

- **Event A (drawing a heart)**: There are 13 hearts in a deck.
- **Event B (drawing a red card)**: There are 26 red cards in a standard deck (13 hearts + 13 diamonds).

The intersection \( A \cap B \) refers to the scenario where a card drawn is both a heart and a red card. Since all hearts are red cards, the number of favorable outcomes for the intersection \( A \cap B \) is the number of hearts, which is 13.

Now, we can find the probability of drawing a heart first from a deck of 52 cards, followed by drawing any red card from the remaining cards.

1. **Probability of drawing a heart first**:
\[
P(A) = \frac{13}{52} = \frac{1}{4}
\]

2. **After drawing a heart**, there will be 51 cards left in the deck (since we do not replace the card) and 25 red cards remaining (13 diamonds + 12 hearts).

3. **Probability of drawing a red card on the second draw**:
\[
P(B | A) = \frac{25}{51}
\]

Now we compute the joint probability \( P(A \cap B) \):
\[
P(A \cap B) = P(A) \cdot P(B | A) = \frac{13}{52} \cdot \frac{25}{51}
\]
\[
= \frac{1}{4} \cdot \frac{25}{51} = \frac{25}{204}
\]

Calculating \( \frac{25}{204} \) gives approximately:
\[
\frac{25}{204} \approx 0.1225
\]

Rounding to the nearest tenths place:
\[
P(A \cap B) \approx 0.1
\]

Thus, the final answer is:
\[
\boxed{0.1}
\]