How do you calculate the probability of a higher flush?

[lunchmeat]

In hold 'em when the board shows four cards of the same suit and you hold a fifth card of that suit how do you determine the odds of an opponent holding a higher flush.

As an example, suppose you hold 9 [img]/images/graemlins/diamond.gif[/img] 9 [img]/images/graemlins/club.gif[/img]. The board is T [img]/images/graemlins/club.gif[/img] 7 [img]/images/graemlins/club.gif[/img] 8 [img]/images/graemlins/heart.gif[/img] 4 [img]/images/graemlins/club.gif[/img] 2 [img]/images/graemlins/club.gif[/img]. Three other players are in the pot and you have no clue as to what they might be holding. What are the odds your flush is best, and how to you come to that answer?

Thanks in advance

[George Rice]

There are 8,145,060 possible combinations of the six cards held in opponents hands (45*44*43*42*41*40/6*5*4*3*2).

There are 3,838,380 possible combinations out of the above combinations that do not contain one of the five cards (A,K,Q,J, or T of clubs) that will beat your nine high flush (40*39*38*37*36*35/6*5*4*3*2).

That leaves 4,306,680 combinations that contain one or more of the five cards that will beat you. 4,306,680/8,145,060 = 52.87% chance that you are beaten, if the other cards are totally random.

In practice, many of the combinations that contain one of the five clubs would have never been played to begin with, or would have folded earlier in the hand, as well as many of the combinations that you could beat (of which there are more of). Let's guess that comes to about a 60-65% chance you are beaten.

But that's not the end of it. What is the liklihood that you are beaten if someone bets? What is the liklihood you are beaten if you bet and someone calls (or raises). What is the liklihood that you can bet and get a higher club to fold? What is the liklihood you can bet and get a worse hand to call? What is the liklihood someone will bluff at the pot or bet a smaller club? You need to consider all of these things and more before deciding how you will handle the betting on the river.

[Copernicus]

Minor correction...the T is out so there are only 4 cards that beat you, so there is a 61% chance of not being beaten using the same formula (although combinations arent really necessary, just string out the next 6 cards 41/45*40/44 etc.)