There are few things that are so unpardonably neglected in our country as poker.
ID: 3057123 • Letter: T
Question
There are few things that are so unpardonably neglected in our country as poker. The upper class knows very little about it. Now and then you find ambassadors who have sort of a general knowledge of the game, but ignorance of the people is fearful. Why, I have known clergymen, good men, kind-hearted, liberal, sincere, and all that, who did not know the meaning of a “flush.” It is enough to make one ashamed of one’s species.
-Mark Twain
Find the probabilities for the following poker hands. They are arranged in decreasing order of probability.
a) Straight flush. (Five cards in a sequence and of the same suit.)
b) Four of a kind. (Four cards of one face value and one other card.)
c) Full house. (Three cards of one face value and two of another face value.)
d) Flush. (Five cards in a sequence. Does not include a straight flush.)
e) Straight. (Five cards in a sequence. Does not include a straight flush. Ace can be high or low.)
f) Three of a Kind. (Three cards of one face value. Does not include four of a kind or full house.)
g) Two pair. (Does not include four of a kind or full house.)
h) One pair. (Does not include any of the aforementioned conditions.)
Explanation / Answer
Solution:-
a) The probability that the straight 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
b) Probability of Four of a kind = 0.000240.
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
Probability of four of a kind = 624/2,598,960 = 0.000240.
c) Probability of getting a full house = 0.001441
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.
d) Probability of flush = 0.00198079.
Here all 5 cards are from the same suit
The number of such hands =4C1 × 13C5 = 5148
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of flush = 5148/2,598,960 = 0.00198079.
e) Percentage of straight is 0.3925%.
Here all 5 cards are from the same suit
The number of such hands = (10C1 × (4C1)5 ) - 40= 10,200
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of straight = 10200/2598960 = 0.003925
Percentage of straight is 0.3925%
f)
Probability of three of a kind = 0.021128.
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
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of three of a kind = 54,912/2,598,960 = 0.021128.
g)
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
h)
Probability of one pair = 0.422569.
This the hand with the pattern AABCD, where A, B, C and D
The number of such hands = 13C1 × 4C2 × 12C3 × (4C1)3 = 1,098,240
Total number of combinations of different hands = 52C5 = 2,598,960
Probability of one pair = 1,098,240/2,598,960 = 0.422569.
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