True Flush Odds?

[Blaze1967]

It is common knowledge that the odds of making your flush by the river after flopping 2 of your suit in Texas Holdem is 1.86 to 1. However, those odds assume that the 9 remaining suited cards are still in the deck. But in a 10 handed game some of those cards will be dealt out and mucked before the flop, and or will be in other players hands that see the flop with you. Has anyone figured out the average percentage of the 13 suited cards being dealt out of the deck in a 10 handed game? Then use the remaining figure instead of 9 to come up with the true odds.

We are taught the the value of playing drawing cards go down in a short handed game but if you think about it less cards are being dealt off the deck in a short handed game thus the true odds of making a flush are closer to the 1.86 to 1 than in a full game.

[Bozeman]

Suppose you have AsTs, and the flop has 2s 6s 9d. On average, there will be 9/47*18=3.45 spades in you opponents hands. So there will be 5.55 spades left in the deck, which has 29 cards. So you should miss on the turn ~81% of the time, and miss on the river ~80% of the time. So you will make your flush ~35% of the time, for ~1.85 odds against. And the disagreement of this result with the familiar result is an error of this method, not that one ...

Craig

[Copernicus]

If you are unable to read anything about the other players hands and their flush possibilites from the action, you do not adjust the 9 downward, whether there you are playing heads up or 11 handed, the odds (probabilities) dont change.

Take the very simple example of drawing on the river, and with only one opponent, and he accidentally flashed one of his cards which wasnt of your suit. There are now 45 unknown cards left, and, as is common knowledge, your chances of making the flush are 9/45, or 20%.

But now, you think, he does have another card which could have my flush card, so how do I adjust for that chance?

His other card is one of your flush cards 9/45 times, which reduces your odds to 8/44. His other card is not your flush card 36/45 times, which increases your odds to 9/44. Totalling the two chances, 9/45*8/44+36/45*9/44=
9/45*(8/44+36/44)=9/45*1...the same 20% you started with.

You will always get the same netting effect no matter how many players there are and cards that are dealt. When they have your card your chances go down, but when they dont your chances go up, and they always exactly offset.

If prior action tells you that some of your outs may be gone (for example your read is that the only hand one of your opponent can possibly have is the same flush draw, but yours is to the nuts) then you can adjust your outs from 9 to 7 for his hand, but disregard all of the other players. Remember though, that if you put him on those two of your suit, the denominator of your probability is also reduced by 2 cards. Similarly if you think there is a 50/50 chance he has two of your flush cards, but the other half of the time he could have anything else other than any of your flush cards (eg half the time you put him on a flush draw in another suit), you can reduce your outs by one card(2 cards gone half the time) and the denominator by 1 card.