Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a spade given that the second card is a club? (Round your answer to three decimal places)
1/4
hmmm. seems this answer is incorrect! :(
To find the probability that the first card is a spade given that the second card is a club, we can use conditional probability.
Step 1: Determine the total number of outcomes.
In a standard deck of 52 cards, there are 52 possible outcomes for the first draw.
Step 2: Determine the number of ways the first card can be a spade.
There are 13 spades in a deck of 52 cards, so there are 13 possible outcomes for the first card being a spade.
Step 3: Determine the number of ways the second card can be a club.
If the first card is a spade, there will be 51 cards left in the deck. Among these, there are 13 clubs. Therefore, there are 13 possible outcomes for the second card being a club, given that the first card is a spade.
Step 4: Calculate the probability using conditional probability formula.
The conditional probability P(A|B), where A is the event that the first card is a spade and B is the event that the second card is a club, can be calculated using the formula:
P(A|B) = P(A and B) / P(B)
We have already determined that there are 13 possible outcomes for the first card being a spade and 13 possible outcomes for the second card being a club given that the first card is a spade.
Therefore, P(A and B) = 13/52 * 13/51 = 169/2652
We also determined that there are 13 possible outcomes for the second card being a club, irrespective of the first card.
Therefore, P(B) = 13/52 = 1/4
Now, let's calculate the probability:
P(A|B) = P(A and B) / P(B)
= (169/2652) / (1/4)
= (169/2652) * (4/1)
= 676/2652
≈ 0.255 (rounded to three decimal places)
Therefore, the probability that the first card is a spade given that the second card is a club is approximately 0.255, or 0.255 to three decimal places.