Question

2 cards are drawn one by one without replacement from a pack of standard 52 cards. find probability that first card is spades and second card is hearts

• A. 13/52
• B. 13/51
• C. 1/4
• D. 13/204

Answer 1

Answer: (D)

13/204

Answer 2

Total number of cards in a standard deck = 52. Number of spades = 13. Number of hearts = 13.

We want to find the probability that the first card drawn is a spade and the second card drawn is a heart, without replacement.

1. Probability of drawing a spade first ($P(S_1)$): There are 13 spades out of 52 cards. $P(S_1) = \frac{13}{52} = \frac{1}{4}$

2. Probability of drawing a heart second, given the first was a spade and not replaced ($P(H_2 | S_1)$): After drawing one spade, there are 51 cards remaining in the deck. The number of hearts remains 13. $P(H_2 | S_1) = \frac{13}{51}$

3. Probability that the first is spades and the second is hearts ($P(S_1 \text{ and } H_2)$): This is the product of the probabilities from step 1 and step 2. $P(S_1 \text{ and } H_2) = P(S_1) \times P(H_2 | S_1)$ $P(S_1 \text{ and } H_2) = \frac{1}{4} \times \frac{13}{51}$ $P(S_1 \text{ and } H_2) = \frac{13}{204}$