Calculations for hands with no wild cards:
The number of ways that 5 cards can be dealt from a deck of 52 cards is 52 C5 = (52!)/(5!47!) = 2,598,960
Of these hands, how many are 4 of a kind? Four of a kind requires 4 cards of the same denomination plus any one of the remaining 48 cards. Because the choice of the extra card is independent from the choice denomination of the other four, the calculation is 13 * 48 = 624. (Alternatively: 13 C1 * 4 C4 * 48 C1 = 13 * 1 * 48 = 624)
The probability of being dealt Four of a Kind from a deck of 52 cards is therefore 624/2,598,960 = 0.000240
How many ways are there to get a full house? A full house requires 3 cards of one denomination and 2 cards of another. The matching triple can be any of 13 the denominations and the pair can be any of the remaining 12 denominations. There are 4 C3 = 4 ways to select the suits of the matching triple and 4 C2 = 6 ways to select the matching pair. As these selections are independent, the calculation for the number of ways to select a full house from a pack of 52 cards is 13 * 4 * 12 * 6 = 3,744.
The probability of being dealt a full house from a pack of 52 cards is therefore 3,744/2,598,960 = 0.001441
The probabilities of being dealt the other hands can be calculated using similar methods:
Three of a kind:
13 C1 * 4 C3 * 12 C2 * 4 C1 * 4 C1 = 13 * 4 * 66 * 4 * 4 = 54912
Two Pair:
13 C2 * 4 C2 * 4 C2
* 44 C1 = 123,552
etc …
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