With explnation please If it is assumed that all (52 5) poker hands are equally
ID: 3201473 • Letter: W
Question
With explnation please
If it is assumed that all (52 5) poker hands are equally likely, what is the probability of being dealt a flush? (all 5 cards are of the same suit) one pair? (the cards have denominations a, a, b, c, d where a, b, c, d are all different) two pairs? (the cards have denominations a, a, b, b, c where a, b, c are all different) three of a kind? (the cards have denominations a, a, a, b, c where a, b, c are all different) four of a kind? (the cards have denominations a, a, a, a, b where a and b all different)Explanation / Answer
Solution:
Total number of combinations of differenet hands = 52C5 = 2,598,960
a) 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 differenet hands = 52C5 = 2,598,960
Probability of flush = 5148/2,598,960 = 0.00198079.
b) 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 differenet hands = 52C5 = 2,598,960
Probability of one pair = 1,098,240/2,598,960 = 0.422569.
c) 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 differenet hands = 52C5 = 2,598,960
Probability of two pairs = 123,552/2,598,960 = 0.047539
d) 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 differenet hands = 52C5 = 2,598,960
Probability of three of a kind = 54,912/2,598,960 = 0.021128.
e) 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 differenet hands = 52C5 = 2,598,960
Probability of four of a kind = 624/2,598,960 = 0.000240.
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