3 flushes

[beta1607]

In a game early there was a case of flush over flush over flush. What are the odds of 3 players making a flush in hold'em assuming there are only 3 suited cards on the river? Also if someone could show the math on how to do this so I dont have to ask these questions in the future I would appreciate it.

[gaming_mouse]

[ QUOTE ]
In a game early there was a case of flush over flush over flush. What are the odds of 3 players making a flush in hold'em assuming there are only 3 suited cards on the river? Also if someone could show the math on how to do this so I dont have to ask these questions in the future I would appreciate it.

[/ QUOTE ]

I'll do the chance of three of more flushes, which will be close to the chance of exactly three anyway. Assume a full ring game. Also, we will ignore the action. That is, we'll simplify the problem as follows:

Ten players are dealt cards. A 5 card board is then dealt out and it contains 3 suited cards. What is the chance of 3 flushes?

First we ask: What is the chance that players in seat 1,2 and 3 are all dealt flushes?

((10 choose 2)/(47 choose 2))*((8 choose 2)/(45 choose 2))*((6 choose 2)/(43 choose 2))

We now multiply by the number of possible ways to group 3 players, (10 choose 3). This will give us a very accurate approximation to the problem in bold, since higher order terms in the inclusion-exclusion formula are small enough to be ignored. Thus the answer is:

(10 choose 3)*((10 choose 2)/(47 choose 2))*((8 choose 2)/(45 choose 2))*((6 choose 2)/(43 choose 2))=0.00234689906

About 1 in 426. Note that the chance of this happening in a game is actually less, since many players will not play any two suited cards (although many do in low limit), and also because many will not see the turn, and thus the times when the flush is made by runner-runner cards should be discounted.

Anyway, for a very loose low limit game, the answer I gave is a decent ballpark figure. Note that this DOES NOT mean it will happen 1 in 500 hands. It means that of the times when the board has a 3-flush on the river, it happens about 1 in 500. 1 in 600 or 700 is probably more accurate tho.

HTH,
gm

[beta1607]

Thanks!