Question: Two cards are drawn in succession from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a diamond
(A) If the cards are drawn without replacement?
(B) If the cards are drawn with replacement?
1. **Problem Statement:**
We want to find the probability that the first card drawn is a spade and the second card drawn is a diamond from a standard 52-card deck.
2. **Key Information:**
- Total cards in deck: $52$
- Number of spades: $13$
- Number of diamonds: $13$
3. **Part (A): Without Replacement**
- The first card is a spade: Probability = $\frac{13}{52} = \frac{1}{4}$
- After drawing a spade, there are now $51$ cards left.
- The second card is a diamond: Probability = $\frac{13}{51}$
4. **Calculate combined probability without replacement:**
$$
P = \frac{13}{52} \times \frac{13}{51} = \frac{1}{4} \times \frac{13}{51} = \frac{13}{204}
$$
5. **Part (B): With Replacement**
- The first card is a spade: Probability = $\frac{13}{52} = \frac{1}{4}$
- Since the card is replaced, the deck is back to $52$ cards.
- The second card is a diamond: Probability = $\frac{13}{52} = \frac{1}{4}$
6. **Calculate combined probability with replacement:**
$$
P = \frac{13}{52} \times \frac{13}{52} = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}
$$
**Final answers:**
- (A) Without replacement: $\frac{13}{204} \approx 0.0637$
- (B) With replacement: $\frac{1}{16} = 0.0625$