Question

In a deck of 52 playing cards, what is the probability of drawing either a heart or a club, then a spade or a diamond, if each card is returned to the deck before drawing the next one?

Answer 1

To solve this problem, we will determine the probability of drawing a heart or a club followed by a spade or a diamond, with the condition that each card is replaced after drawing.

1. **Probability of drawing a heart or a club:**
- In a standard deck of 52 cards, there are 13 hearts and 13 clubs.
- Thus, the total number of hearts and clubs combined is:
\[
13 + 13 = 26
\]
- Therefore, the probability of drawing a heart or a club (\(P(H \cup C)\)) is:
\[
P(H \cup C) = \frac{26}{52} = \frac{1}{2}
\]

2. **Probability of drawing a spade or a diamond:**
- There are also 13 spades and 13 diamonds in the deck.
- Thus, the total number of spades and diamonds combined is:
\[
13 + 13 = 26
\]
- Therefore, the probability of drawing a spade or a diamond (\(P(S \cup D)\)) is:
\[
P(S \cup D) = \frac{26}{52} = \frac{1}{2}
\]

3. **Calculating the overall probability:**
- Since each draw is independent (because the card is replaced), the probability of both events happening in sequence is the product of their individual probabilities:
\[
P(H \cup C \text{ and } S \cup D) = P(H \cup C) \times P(S \cup D) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]

Thus, the probability of drawing either a heart or a club followed by a spade or a diamond, with each card replaced, is:

\[
\boxed{\frac{1}{4}}
\]