What is the probability of being dealt the following poker hands (5 cards out of
ID: 3305548 • Letter: W
Question
What is the probability of being dealt the following poker hands (5 cards out of a standard deck of 52 cards):
Please give the detailed soulution of this quaestion.
a. a royal flush-lo, J, Q, K, A all in the same suit (e.g., 10 Jolo» 1K4 A+) b. a royal flush-1o, J, Q, K, A all in the same suit (e.g., 10.J.IQ.K IA.) c. a straight flush (including royal flushes)-five cards in order, all in the same suit (e.g., 80194/10-J+) d. a straight-five cards in order, but they don't have to be in the same suit (e.g., 7 e. a full house-three of a kind and a pair (e.g., 7 17 17A10 10) f. a flush-five cards of the same suit eg, 1015 10o*9)Explanation / Answer
Solution:-
a) The probability that the hand is a royal flush is 0.00000154.
Total combination of royal flush = 4
Total number of combinations of different hands = 52C5 = 2,598,960
The probability that the hand is a royal flush is = 4/2598960 = 0.00000154.
b)
The probability that the hand is a royal flush is 0.00000154.
Total combination of royal flush = 4
Total number of combinations of different hands = 52C5 = 2,598,960
The probability that the hand is a royal flush is = 4/2598960 = 0.00000154.
c) 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
d) The probability that the hand is straight is 0.003925.
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
e) 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.
f) 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.
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