Robb's 3000th Post

Ok Robb, some flop probabilities please. How often do we see:

- Rainbow, two tone, mono flops?
- Paired flops?
- 3 of a kind flops?
- A high?
- A or K high?
- A, K or Q high?

ta

Originally Posted by Lucothefish

- Rainbow, two tone, mono flops?ta

There are "52 choose 3" = 22,100 flops, if we ignore any hole cards, as if we are railing.

Monotone: 4c1 * 13c3 / 52c3 = 1144 / 22100 ~ 5.18%
Two tone: (4c1 * 13c2 * 3c1 * 13c1) = 12168 / 22100 ~ 55.06%
Rainbow: 4c3 * ( 13c1 )^3 = 8788 / 22100 ~ 39.76%

If we take into account all of Hero's possible hole cards, pairs, unsuited, and suited, these probabilities are exactly the same. But it's a bunch more complicated to do the math.

Originally Posted by Lucothefish

- Paired flops?
- 3 of a kind flops?

Watching the game from the rails, not knowing any hole cards, here are the probabilities:

Trips: 13c1 * 4c3 = 52 / 22100 ~ .24%
Paired: 13c1 * 4c2 * 13c1 * 4c1 = 3744 / 22100 ~ 16.94%
3 diff values: 13c3 * (4c1)^3 = 18304 / 22100 ~ 82.82%

Fortunately, the probabilities add up to 100% and account for all the possibilities.

Originally Posted by Lucothefish

- A high?
- A or K high?
- A, K or Q high?

There are several ways to answer this question. We hate Ace high flops the most when we hold KK or hands like KQ. These are most easily accomplished by using the set complement, and calculating the probability of it NOT happening, then subtracting from 100%.

Probability of at least one Ace on flop, given we have KK:

P = 1 - 46c3 / 50c3 ~ 31.31%

Probability of at least one Ace or King, given we have QQ:

P = 1 - 42c3 / 50c3 ~ 48.05%

Probability of at least once Ace, King or Queen, given we have JJ:

P = 1 - 38c3 / 50c3 ~ 61.83%

Probability of at least one overcard, given we have TT:

P = 1 - 34c3 / 50c3 ~ 72.92%

Probability of at least one overcard, given we have 99:

P = 1 - 30c3 / 50c3 ~ 81.63%


If you have some other scenarios you'd like the counts on, like number of times Hero holds Ax and hits an Ace-high flop, let me know.

Awesome, thanks!