Just a thought: so it was 48-1 to hit your two pair...how about the odds of hitting two pair for two players? I leave this as an exercise for the student.
There are 3 different board combinations that can give you 2 pair on the flop using both your hole cards. Either the 1st & 2nd flop card pairs your pocket cards and the last one doesn't, the 1st & 3rd flop cards match your pocket cards and the 2nd doesn't, or the 2nd & 3rd flop cards match your pocket cards and the 1st one doesn't. So I calculated the odds of each happeneing and added them all together.
Scenario #1. The chances of the 1st flop card hitting your pocket cards is 6 out of 50. The chances of the second flop card hitting your OTHER pocket card (given the fact that the first one did) is 3 out of 49. The chances that the third one will NOT hit your pocket cards (given the fact the the first two did) is 44 out of 48. Therefore the chances that the 1st & 2nd flop cards will match each of your pocket cards while the 3rd one doesn't is (6/50) * (3/49) * (44/48).
Scaenario #2. The chances of the 1st fop card hitting your pocket cards is 6 out of 50. The chances of the second card not hitting your pocket cards (given that the first flop card did) is 44/49. The chances of the third flop card hitting your OTHER pocket card is 3/48. Therefore, the chances that the 1st and 3rd flop cards will match each of your pocket cards while the second one doesn't is (6/50) * (44/49) * (3/48).
Scenario #3. The chances of the first card not matching your pocket cards is 44/50. The chances of the 2nd flop card hitting you (given the fact that the first one didn't) is 6/49. The chances of the 3rd flop card hitting your other pocket card is 3/48. Therefore, the chances of the 2nd & 3rd flop cards hitting each of your pocket cards is (44/50) * (6/49) * (3/48).
If we add all 3 scenarios together we get: ((6/50) * (3/49) * (44/48)) + ((6/50) * (44/49) * (3/48)) + ((44/50) * (6/49) * (3/48)) = .02020 or 2.02%
You could use the same logic to caculate the chance that some flop 2 pair on the flop - you would only care if you had second 2 pair.
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