Probability and Four of a Kind
It would seem that 4 of a kind would be harder to get than a straight flush. After all there are only 13 different numbered cards in a deck. Yet it is more likely that you should be dealt 4 of a kind than a straight flush. The reason is, you only need 4 cards to make up four of a kind, and you have five chances to get them.
The actual odds of getting 4 of a kind are one in 4165. This can be obtained by figuring that there are 624 combinations of cards that make up four of a kind out of 2,598,960 possible hands. How do we derive these 624 combinations?
13 X 48 = 624
Pretty simple really, 13 is the different numbered cards while the 48 is the number of other cards available after 4 of a kind take up the first four positions of a 5 card poker hand.
If you already have three of a kind your chances of achieving four of a kind after discarding 2 cards improves markedly. This means that if you already have 3 of a kind, the wise move is to shoot for four of a kind. Complete failure still leaves you with a tolerably good hand and the 2/47 chances that you will get your fourth card are pretty good (considering). There is also the possibility of a making three of a kind into a full house. The chances of doing so (drawing two cards and already holding three of a kind) can be derived using the following calculation:
46/47 X 3/46 = 138/2162 = approximately 1/15.667
Where did these numbers come from? The denominator of 46/47 is the number of cards left in the deck. The 46 represents the number of those cards that would allow you to get two of a kind. (We have to allow for the fact that there is one card which you might draw from which it would be impossible to get two of a kind - that would be one of the three of a kind that you already have in your possession - yet drawing this card would be a good thing!) The denominator of 3/46 is the number of cards remaining in the deck after your first draw. The 3 is the number of chances of getting the same card as your first draw. The result, one chance in 16, is pretty good for getting your full house. Combined with the opportunity to draw the 4th card of your 3 of a kind we have pretty good chances for this hand.
2/47 + 1/15.667 = approximately 1/10.5
Just over ten percent is pretty good odds on markedly improving an already good hand.