ChanceAndGames Joined: Feb 8, 2014
• Posts: 1
February 8th, 2014 at 12:50:33 AM permalink
In five card poker, if you have a flush with a high of 7, and then the dealer deals another five cards to another player, what are the odds that he has a flush that beats yours?
paisiello Joined: Oct 30, 2011
• Posts: 546
February 8th, 2014 at 3:29:50 AM permalink
Probability of a flush or a straight flush:

[C(8,5)+C(13,5)*C(3,1)] / C(47,5) = [(56)+(1,287)(3)] / ( 1,533,939 )
= 3,917 / 1,533,939
= 0.2554%

Now since only 3 of the flushes will be less then a 7 high flush without being a straight flush:

(3,917 - 3) / 1,533,939 = 0.2552%
JB Joined: Oct 14, 2009
• Posts: 2087
February 8th, 2014 at 5:04:14 AM permalink
Quote: ChanceAndGames

In five card poker, if you have a flush with a high of 7, and then the dealer deals another five cards to another player, what are the odds that he has a flush that beats yours?

It depends on the specific ranks in your flush.

If your flush is 23457, the probability of being beaten by another flush is 3881/1533939 = 0.002530 = about 1 in 395.24.
If your flush is 23467, the probability of being beaten by another flush is 3878/1533939 = 0.002528 = about 1 in 395.55.
If your flush is 23567, the probability of being beaten by another flush is 3875/1533939 = 0.002526 = about 1 in 395.86.
If your flush is 24567, the probability of being beaten by another flush is 3872/1533939 = 0.002524 = about 1 in 396.16.

Paisiello's answer was close, but didn't take everything into account.

First, let's consider the other suits. There are 3 of them, so however many flushes we come up with in each suit, we'll have to multiply that quantity by 3. Of the combin(13,5) ways to select five cards of that suit, 10 of them are a straight flush, so we must subtract those out. We also must subtract out the quantity of flushes which do not beat yours (they either tie or lose to it). This is why the specific ranks in your flush matter. So this portion of the formula is:
combin(3,1) * (combin(13,5) - 10 - n), where n is the number of rank sequences which would not beat your flush.

If your flush is 23457, n is 1 because there is 1 rank sequence (23457) which does not beat your flush.
If your flush is 23467, n is 2 because there are 2 rank sequences (23457 and 23467) which do not beat your flush.
If your flush is 23567, n is 3 because there are 3 rank sequences (23457, 23467, and 23567) which do not beat your flush.
If your flush is 24567, n is 4 because there are 4 rank sequences (23457, 23467, 23567, and 24567) which do not beat your flush.

Next, we'll consider the same suit. There are 8 cards left in it. Since your flush is 7-high, it is impossible for the other player to form a 5-high, 6-high, 7-high, 8-high, 9-high, 10-high, or Jack-high straight flush. This leaves 3 possible straight flushes he could have, which we must subtract out. So there are (combin(8,5) - 3) possible flushes in the same suit, all of which would of course beat your flush. We need to add this total to the total we came up with for the other 3 suits.

Then we divide the sum by the number of combinations of 5-card hands the other player could have, which is combin(47,5).

The final formula is: ((combin(3,1) * (combin(13,5) - 10 - n)) + (combin(8,5) - 3)) / combin(47, 5)
where n is the number of flushes which tie or lose to your flush.