Question

Find the probability of the following five-card poker hands from a 52-card deck. In poker, aces are either high or low.

Four of a kind (4 cards of the same value)

I keep getting 1/199,920, but that's not one of my options.

Answer 1

Suppose you were after the probability of getting four aces in five cards. Suppose also that the four aces were the first four cards out of the five you were dealt. The chances of getting that would be (4/52)x(3/51)x(2/50)x(1/49).

But the 5th card could be any of the five you are dealt, so the above needs to be multiplied by 5. Also, your four of a kind could be any of the 13 different face-values, so multiply the above by 13 to get all of them. That would give you

(13x5x4x3x2)/(52x51x50x49) = 1/4165.
Is that one of your options?