How do I find the number of 3-of-a-kind hands possible in poker (without and with 1 wild card)?

Question

I'm trying to figure the probability of each hand of poker; then the probablility of each hand of poker using a joker as a wild card. the hands are: Straight flush, 4 of a kind, full house, flush, straight, 3 of a kind, 2 pair, 1 pair. I think I have it for the first four hands, but I'm stuck on the 3 of a kind.

Answer 1

You have to be kidding Judy. There are 14 set of 4 numbered cards, You can make 14 sets of 3 of a kind. With only 1 wild card you can only make i pair more. I have never heard of a game with only 1 wild card, as even a joker come 2 to a pack. but saying that I do not remember how many "man with the axe" there are. I do not have a deck close by to check.

Answer 2

Depending on what type of poker you play.

Assuming 5 card stud.

# type of cards: 13

prob of getting a specific three of a kind:

(4/52)(3/51)(2/50)(48/49)(44/48)

making sense of this:

(4/52)(3/51)(2/50): Prob of getting 3 specific cards of the same type (king, jack, 9, etc)

(48/49): Prob of getting any other card besides the 4th (no quads)

(44/48): Prob of getting any other card besides the 4th (no quads) and a pair to the previous card drawn (No boat)

Prob of achieving that specific hand:

(4/52)(3/51)(2/50)(48/49)(44/48)

Now we want to figure out all possible ways for that hand, (instead of KKK29 or JJJ35, we want 2J3JJ etc)

thats just 5!/3! (harder to explain this quickly)

so for any specific card, the probability is:

(4/52)(3/51)(2/50)(48/49)(44/48)*5!/3!

for all cards, multiply by 13

13*(5!/3!)*(4/52)(3/51)(2/50)(48/49)(4...

since 5!=1*2*3*4*5, 3*=1*2*3

prob= 13*4*5*(4/52)(3/51)(2/50)(48/49)(44/48)