6. In a poker game, 5 cards are dealt from a standard 52 card deck that has been
ID: 3042489 • Letter: 6
Question
6. In a poker game, 5 cards are dealt from a standard 52 card deck that has been well shuffled. You are the only player in this scenario. (Note: if you are not familiar with poker hands, you may want to look up what some of these are online—also check out Chapter 23 in the textbook.)
b) What is the probability that you are dealt two pairs?
c) What is the probability that you are dealt a three of a kind or 4 of a kind? (Note: three of a kind means that the other 2 cards are NOT a pair.)
d) What is the probability that you are dealt a full house? (Note: a full house means 3 of a kind and a pair.)
e) What is the probability that you are dealt a flush, three of a kind, or 4 of a kind? (any type of flush is acceptable)
Explanation / Answer
Solution:-
b) Probability of two pairs = 0.047539
This hand has the pattern AABBC where A, B, and C are from distinct kinds.
The number of such hands = 13C2 × 4C2 × 4C2 ×11C1 × 4C1 = 123,552
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of two pairs = 123,552/2,598,960 = 0.047539
c) Probability of three of a kind or Four of a kind = 0.021368.
This hand has the pattern AAABC where A, B, and C are from distinct kinds.
The number of such hands = 13C1 × 4C3 ×12C2 × (4C1)2 = 54,912
This hand has the pattern AAAAB where A and B are from distinct kinds.
The number of such hands= 13C1 × 4C4 × 12C1 × 4C1 = 624
Total number of combinations of different hands = 52C5 = 2,598,960
The number of such hands three of a kind or Four of a kind = 55,536
Probability of three of a kind or Four of a kind = 55,536/2,598,960 = 0.021368.
d)
This hand has the pattern AAABB where A and B are from distinct kinds.
The number of such hands = 13C1 × 4C3 × 12C1 × 4C2. = 3774
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of getting a full house = 3774/2,598,960 = 0.001441
e)
The probability that the flush including royal flush is 0.0000154.
Each straight flush is uniquely determined by its highest-ranking card. These ranks go from 5 (A-2-3-4-5) up to A (10-J-Q-K-A) in each of the 4 suits.
The number of such hands = 36 + 4 = 40
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of straight flush including royal flush is = 40/2598960 = 0.0000154
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