When the flop is monotone(question for aloiz or another math wiz)

[Vannek]

Aloiz, I remember a question about trips where you said
that the chance someone has a trip when the board is paired is ~7%(not an exact approximation) per opponent, do you have any similar rule of thumb when the flop is monotone?

[slickpoppa]

You don't really need to be a math wiz to answer this one. Think about from the perspective of one player. 3 cards of the same suit have been dealt. There are 49 unknown cards left on the deck, and 10 of them are of the relevant suit. So the prob of being dealt just one of those is 10/49. The prob of being dealt another after than is 9/48. Mutlplying these together gives u the prob of you being dealt that flush, assuming the monotone flop.
(10/49)*(9/48) = .0382 or 3.82%
So thats the prob of any one player having that flush.

[Vannek]

So for two people it would be 3.82 * 2?

[JFB37]

[ QUOTE ]
So for two people it would be 3.82 * 2?

[/ QUOTE ]

Not even close. 3.82 * 2 = 7.64.

Think about it. The possibility of two people having the flush is not going to be higher than one person having the flush.

Continue the logic used above. The probability of two specific people having the flush is 10/49 x 9/48 x 8/47 x 7/46 = 0.001 or 0.1%.

[Vannek]

JF,
I mean at least of them having the flush

[mmbt0ne]

Probability that one has the flush is:
2(Probability one has flush) - Probability they both have a flush.
I'm too tired to use numbers, I hope that explains it.

[Precision1C]

With a monotone flop column one is the chance NOBODY has a flush when you have no matching card in your hand with the listed number of opponents, column 2 is if you have one matching card and column 3 is if you have a flush yourself.
0 1 2
9 68.2% 73.7% 79.0%
8 71.2 76.3 81.0
7 74.2 78.9 83.2
6 77.5 81.6 85.4
5 80.8 84.4 87.7
4 84.4 87.3 90.0
3 88.0 90.3 92.4
2 91.8 93.3 94.9
1 95.8 96.7 97.4

[BruceZ]

[ QUOTE ]
Aloiz, I remember a question about trips where you said
that the chance someone has a trip when the board is paired is ~7%(not an exact approximation) per opponent, do you have any similar rule of thumb when the flop is monotone?

[/ QUOTE ]

What you want is in the table at the bottom of this post. It gives the probability of one or more opponents having been dealt 2 flush cards given 2,3,4, or 5 flush cards exposed on the flop, and for 1-10 opponents. So if there are 3 flush cards on the board, and you hold none in your hand, you would look at the column for 3 exposed flush cards, and find that the probability is 33.4% that at least one of your 9 opponents has a flush. Note that this assumes that your opponents hold random hands, and in a real game the probabilities will be affected by the hands your opponents play.

As for a rule of thumb, it looks like it starts out about 4% per opponent for shorthanded, and goes to about 3% per opponent for a full table. In this case, simply multiplying the result for 1 opponent by 9 is not a good approximation. I computed these using the inclusion-exclusion principle.