What is the probability of getting a 4-of-a-kind? Daina

Answer

In a simple draw it is 0.024%, but there are a few different answers depending on what game you are playing. I'll go through a few of the common scenarios. First though I need to note that there are 13 basic four of a kinds, one for each rank in the deck.

For a straight 5 card draw we need to calculate how many will contain four of a kind compared to how many exist in total. Here can use combinations. There are (52c5) = 2598960 total combinations. To make four of a kind we need any one of those 13 four card combinations we mentioned before plus one extra card to make a five card hand. Recall the poker hand is always five cards. After taking out the four cards we want, there are 48 cards left in the deck. Thus there are actually 13 * 48 = 624 four of a kind hands. The chance of getting this on a straight deal is then 624 / 2598960 ~= 0.024%.

In Texas Hold'em the situation changes only a little. You can simply do the same calculations but with 7 cards as opposed to 5. That results in (13*(48c3)) / (52c7) ~= 0.16%. Note that first part of that calculation (48c3) is the number of 3-card combinations from the 48 cards left in the deck (once you remove the four of a kind).

If you already have a pocket pair and would like to know your chance of getting a four of a kind, the calculation is again similar. Simply count the number of community cards which would contain your remaining two cards divided by the total possible combinations. So with a pocket pair you have a (48c3) / (50c5) ~= 0.92% chance. Note we don't multiple by 13 here since only one specific rank is of interest now.