Question

The probability of getting a heart on the first draw is Response area. After replacement, the probability of getting an ace on the second draw is Response area. The probability of getting a heart and an ace in two draws (with replacement) is Response area.

Answer 1

Let's break down the probabilities step by step for a standard deck of cards, which contains 52 cards: 13 hearts, 4 aces (with 1 of those being a heart), and 39 other cards.

1. **Probability of getting a heart on the first draw:**
There are 13 hearts in a 52-card deck. Therefore, the probability \( P(\text{heart}) \) is:
\[
P(\text{heart}) = \frac{13}{52} = \frac{1}{4}
\]

2. **Probability of getting an ace on the second draw (after replacement):**
There are 4 aces in a 52-card deck. Therefore, the probability \( P(\text{ace}) \) is:
\[
P(\text{ace}) = \frac{4}{52} = \frac{1}{13}
\]

3. **Probability of getting a heart and an ace in two draws (with replacement):**
Since the draws are independent (due to replacement), the probability \( P(\text{heart and ace}) \) can be calculated by multiplying the probabilities of the two independent events:
\[
P(\text{heart and ace}) = P(\text{heart}) \times P(\text{ace}) = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52}
\]

So, to summarize:
- The probability of getting a heart on the first draw: \( \frac{1}{4} \)
- The probability of getting an ace on the second draw: \( \frac{1}{13} \)
- The probability of getting a heart and an ace in two draws (with replacement): \( \frac{1}{52} \)