Suppose you’re playing Texas Hold ’Em Poker, and you are dealt a hand witha Jack of Clubs and a Queen of Hearts. The river of 5 cards hasn’t been dealtyet, and you’re hoping to get a straight (5 cards with consecutive ranks, such as10,J,Q,K,A or 8,9,10,J,Q). What is the probability that you will get a straightthat uses both the Jack and Queen in your hand? (This last part is to keep theproblem simple, since it’s possible to get a straight that uses neither card in your hand, suchas A,2,3,4,5, all in the river. Besides, in those cases, ALL players would have a straight, sothat’s not so good for you!)Guide: Our goal is to calculate:P(straight using both cards in hand) = # 5-card combinations that get us a straight with the J & Q# 5-card combinations overall .(a) We’ll start with the denominator, which can be done with a singleformula from class: How many 5-card combinations are possible for theriver? (Keep in mind what is in your hand. Note that how many opponents you haveand what is in their hands is irrelevant since their cards are unknown to you. If it helps,you can think of your opponents’ cards as being indistinguishable from the remainingcards in the deck.) (b) For each of the four steps, how many different ways can we choose? (c) How many different 5-card rivers will give us the cards we need toget a straight using both cards in our hand? (d) Answer the original question: what is the probability that we get a straightthat uses both the Jack of Clubs and Queen of Hearts in our hand?
Suppose you’re playing Texas Hold ’Em Poker, and you are dealt a hand with
a Jack of Clubs and a Queen of Hearts. The river of 5 cards hasn’t been dealt
yet, and you’re hoping to get a straight (5 cards with consecutive ranks, such as
10,J,Q,K,A or 8,9,10,J,Q). What is the probability that you will get a straight
that uses both the Jack and Queen in your hand? (This last part is to keep the
problem simple, since it’s possible to get a straight that uses neither card in your hand, such
as A,2,3,4,5, all in the river. Besides, in those cases, ALL players would have a straight, so
that’s not so good for you!)
Guide: Our goal is to calculate:
P(straight using both cards in hand) = # 5-card combinations that get us a straight with the J & Q
# 5-card combinations overall .
(a) We’ll start with the denominator, which can be done with a single
formula from class: How many 5-card combinations are possible for the
river? (Keep in mind what is in your hand. Note that how many opponents you have
and what is in their hands is irrelevant since their cards are unknown to you. If it helps,
you can think of your opponents’ cards as being indistinguishable from the remaining
cards in the deck.)
(b) For each of the four steps, how many different ways can we choose?
(c) How many different 5-card rivers will give us the cards we need to
get a straight using both cards in our hand?
(d) Answer the original question: what is the probability that we get a straight
that uses both the Jack of Clubs and Queen of Hearts in our hand?
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