Question

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.)

Answer 1

A. P("JQK" | "ACE")=0.2352

Explanation:

A. The definition of conditional probability for events in this case tell us that:

P("JQK" | "ACE")= P("JQK" ∩ "ACE")/P("ACE")

Where P("JQK") is the probability that the first card is a face card and P("ACE") is the probability that the second card is an ace.

Now we need to find P("JQK" ∩ "ACE") and P("ACE") to get what we are asked for.

P("ACE")=(48/52)(4/51)+(4/52)(3/51)

Because if the first card is not an ace, there will be 4 to choose in the second chance but if the first one is an ace, we will have 3 left to choose in the second chance. If we draw a tree diagram for this experiment, the cases above would be on different branches, so we add them.

P("ACE")=0.07692

And for P("JQK" ∩ "ACE"), we have the following:

P("JQK" ∩ "ACE")=(12/52)(4/51)

Because the first card must be a face card, it could not be an ace and after choosing one of the 12 face cards from the deck, there´s 4 aces left to take in the second chance.

P("JQK" ∩ "ACE")= 0.01809

Finally, we can use the definition of the beginning to find our answer:

P("JQK" | "ACE")= P("JQK" ∩ "ACE")/P("ACE")

P("JQK" | "ACE")= 0.07692 / 0.01809

P("JQK" | "ACE")= 0.2352