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 face card (jack, queen, or king) given that the second card is an ace? (Round your answer to three decimal places.)

There a total of 12 face cards in a stack of 52 playing cards. You know that the second card is an ace, so the value of 51 cards are unknown.

Therefore, the probability is 12/51

48/52+44/52

To calculate the probability that the first card drawn is a face card given that the second card is an ace, we can use conditional probability. Let's break down the problem step by step:

Step 1: Determine the probability of drawing an ace as the second card.
There are 4 aces in a standard deck of 52 cards. After one card has been drawn, there are 51 cards remaining, and 4 of them are aces. Therefore, the probability of drawing an ace as the second card is 4/51.

Step 2: Determine the probability of drawing a face card as the first card.
There are 12 face cards in a standard deck of 52 cards (4 jacks, 4 queens, and 4 kings). After one card has been drawn, there are 51 cards remaining, and 12 of them are face cards. Therefore, the probability of drawing a face card as the first card is 12/51.

Step 3: Calculate the conditional probability.
The conditional probability is calculated by dividing the probability of both events happening (drawing a face card and an ace) by the probability of the second event occurring (drawing an ace).

Conditional probability = (Probability of drawing a face card and an ace) / (Probability of drawing an ace)
= (12/51) / (4/51)
= 12/4

Step 4: Simplify the fraction and round the answer to three decimal places.
Conditional probability = 3

Therefore, the probability that the first card is a face card given that the second card is an ace is 3 (or 3/1 when expressed as a fraction), rounded to three decimal places.

To find the probability that the first card drawn is a face card given that the second card is an ace, we can use conditional probability.

First, let's find the probability of drawing an ace on the second card. There are a total of 52 cards in the deck, and after the first card is drawn, there are 51 cards remaining. Since there are 4 aces in the deck, the probability of drawing an ace on the second card is 4/51.

Next, let's find the probability of drawing a face card on the first card. There are 12 face cards in the deck (3 jacks, 3 queens, and 3 kings) out of the total 52 cards. Therefore, the probability of drawing a face card on the first card is 12/52 or 3/13.

Now, we can apply conditional probability. The probability of the first card being a face card given that the second card is an ace can be calculated using the formula:

P(A|B) = P(A ∩ B) / P(B)

Where A represents the event that the first card is a face card, and B represents the event that the second card is an ace.

P(A ∩ B), the probability of both events occurring, is the probability of drawing both a face card and an ace. Since these two events are independent (the outcome of the first card does not affect the outcome of the second card due to drawing without replacement), the probability of both occurring is the product of their individual probabilities:

P(A ∩ B) = (3/13) * (4/51) = 12/663

P(B), the probability of the second card being an ace, is already calculated as 4/51.

Finally, we can plug the values into the formula to calculate the conditional probability:

P(A|B) = (12/663) / (4/51) ≈ 0.067

Therefore, the probability that the first card is a face card given that the second card is an ace is approximately 0.067 or rounded to three decimal places.