Question
What is the probability of being dealt four cards of a kind in a five card hand?
I know it is 0.00024, but I need this as a fraction and I don't know how to do that.
I know it is 0.00024, but I need this as a fraction and I don't know how to do that.
Answers
Reiny
A four-of-a-kind is four cards showing the same number plus any other card.
If we order the 5-card hand with the four-of-a-kind first, we have C(13,1) choices for the number showing on the first four cards. Since we will have all four suits, we have only C(4,4) = 1 way to choose the suits. The
remaining card will be any of the 48 remaining cards:
# 4-of-a-kinds = C(13,1)*C(4,4)*C(48,1) = 13·1·48 = 624
Number of possible 5 cards from 52 = C(52,5) = 2598960
Dividing by the number of possible hands gives the probability:
P(4 - of - a - kind) = 624/2,598,960 = .000240096
If we order the 5-card hand with the four-of-a-kind first, we have C(13,1) choices for the number showing on the first four cards. Since we will have all four suits, we have only C(4,4) = 1 way to choose the suits. The
remaining card will be any of the 48 remaining cards:
# 4-of-a-kinds = C(13,1)*C(4,4)*C(48,1) = 13·1·48 = 624
Number of possible 5 cards from 52 = C(52,5) = 2598960
Dividing by the number of possible hands gives the probability:
P(4 - of - a - kind) = 624/2,598,960 = .000240096