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My hand is any 2 cards that do not have the same rank. What is the probability of flopping a four of a kind?

The formula I used is:

(2 c 1)(3 c 3) / (50 c 3)

c = chooses

2 c 1 - We have 2 ranks in our hand and we need to choose 1 of them
3 c 3 - There are 3 cards of one of the ranks in our hand and need all 3 of them
50 c 3 - Our deck has 50 cards and we draw 3 of them

2 Answers 2

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Your formula is correct, your reasoning is sound. If your hole cards' ranks are different there are exactly two triplets that could complete a four of a kind among all the possible 50 choose 3 triplets the flop is drawn from. The probability is thus 2 / (50 choose 3), which approximately equals 0.009050 %.

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Not a math expert but I do it this way:

(6/50) * (2/49) * (1/48)

(6/50) is hitting a pair with the first flop card. Six cards in the deck will allow you to do so, given two cards of different ranks. (2/49) is then hitting three of a kind with the second flop card. (1/48) is then hitting quads with the third flop card.

Multiply the result by 100 and you get the probability: 0.010204081% = 1 in 9800.

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