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.

1. **Probability of getting a heart on the first draw:**
- A standard deck of cards has 52 cards in total, with 13 of them being hearts.
- Therefore, the probability of drawing a heart on the first draw 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 standard deck of cards.
- Since we draw with replacement, the total number of cards remains 52 for the second draw.
- Therefore, the probability of drawing an 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), we can multiply the probabilities of the two events:
- The combined probability is:
\[
P(\text{Heart and Ace}) = P(\text{Heart}) \times P(\text{Ace}) = \left(\frac{1}{4}\right) \times \left(\frac{1}{13}\right) = \frac{1}{52}
\]

In summary:
- Probability of getting a heart on the first draw: \(\frac{1}{4}\)
- Probability of getting an ace on the second draw: \(\frac{1}{13}\)
- Probability of getting a heart and an ace in two draws (with replacement): \(\frac{1}{52}\)