Originally posted by: dullard
3 of a kind with three cards
1) First card doesn't matter. Whatever number the card is, will determine what the other two must be.
2) 51 cards remain. Three of them match your first card. Thus your odds are 3/51 to get the first two to match.
3) 50 cards remain. Two of them match your first two cards. Thus your odds are 2/50 to get the third card to match.
Overall odds: 3/51 * 2/50 = 1 out of 425.
4 of a kind with five cards
1) There are 672 possible four of a kind hands
2) There are a total of 2,598,960 possible hands.
3) 672/2,598,960 = 1 out of 4165 odds of getting a four of a kind.
Using the same reasoning, for 4 of a kind in 5 cards:
If we get the 4 cards of the same value first, and the other card last, then the probability is 52/52 * 3/51 * 2/50 * 1/49 * 48/48 = 1/20825. But the other card could be first, second, third, fourth, or last. So the probability is 5/20825 = 1/4165.
Getting a three of a kind is almost ten times more likely to happen.
Edit: I'm slow.. clicked quote before I saw his edit.