You are given five cards randomly from a full deck of cards

You are given five cards randomly from a full deck of cards plus a joker. The joker is wild (a joker can be equal to whatever regular card that would create a three of a kind hand even if a better poker hand could be had). What is The probability of three of a kind hand?

Solution

With one joker the hand must consist of a natural pair and two other cards of different rank from each other and from the pair. There are 13 ranks, 4C2 pairs per rank, and one joker. The last two cards have 48 and 44 choices due to removing the four cards of each rank from the remaining cards. To remove the permutations and obtain only combinations I must divide by 2!. The total number of three of a kind hand is then 13 * 4C2 * 1 * (48 * 44)/2! = 164,736 with one joker.

In five-card stud each player is dealt five cards to make the best five-card hand possible. Since there are 52 cards in the deck, then there are 52C5 = 2,598,960 possible combinations of five-card hands possible.

So the probability of three of a kind hand is = 164736/2598960= 0.0063385