What is the chance to get three of a kind if you're holding a pocket pair? It is just under 12% and it is one of the easier probabilities to calculate.

How to calculate

There is usually more than one way to calculate the probability of a given event. Sometimes there is a clear easy way, while other times every way is just difficult. Often one's first intuition on how to approach a problem is the difficult way.

In this case, given pocket pairs what is the chance of hitting the third card of that rank on the flop. A direct question would be, "what is the chance to get a matching rank on the first card, or the second card, or the third card?" Dealing with or statements between dependent events is however a rather tricky situation.

Instead the opposite result can be queried. "What is the chance the third of the rank is not matched on the flop?" This requires that the first card is not a match and the second card is not a match and the third card is not a match. This is a lot easier to calculate as the probability is simply the probability of each event multiplied together.

There are 2 matching cards for a given pocket pair. There are 50 cards left in the deck. 48 cards do not match. The first card has a 48 in 50 chance of not matching. Once that card is gone there is a 47 in 49 chance that the second card doesn't match and a 46 in 48 chance for the third card. A miss on all three cards has the total chance of 48/50 * 47/49 * 46/48 = 88.2%. A simple subtraction then yields the chance of a match 100% - 88.2% = 11.8%.

Nitpickers and Four of a kind

The previous calculation also includes any flop where both remaining matches come up. Generally nobody would be upset if they get four of a kind instead of a set so using the 12% value is perfectly okay. Though for completeness sake that chance of four of a kind should be separated.

Again here it is possible to construct a very complex series of dependent events with or and and which would yield the correct result, but it is error prone and difficult. Instead, since the number for three or four of a kind is already known, to get the strict three of a kind value simply requires calculated the four of a kind chance and subtracting.

Rather than worry about dependent events this can be expressed using outcomes. There are 50 cards left, of which 3 will be selected. That is 19600 combinations. If both matching ranks are taken out, as required cards, there are only 48 cards remaining, so only 48 combinations contain both of the matching cards. Thus 48 / 19600 = 0.24% of getting four of a kind.

Subtracting from the previous chance 11.8% - 0.24% = 11.5% chance of getting strictly three of kind. Not enough of a difference to worry about.

Ooops. 11.8% - 0.24% = 11.6% not 11.5%. The 11.5 comes from using all the unrounded values in the previous calculations.

Excessive nitpicking

To go even further in this direction one would notice that the first calculation also did not exclude a full house, which is also possible with pocket pairs on the flop. That'll just be ignored at this point.