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.
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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: \[ P(H \cup C) = \frac{26}{52} = \frac{1}{2} \]
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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: \[ P(S \cup D) = \frac{26}{52} = \frac{1}{2} \]
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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:
\[ \boxed{\frac{1}{4}} \]
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