a newcomer's query - HE folds effecting winning chances

[clay1m]

I have a question along the same lines as the one posed about AK vs 77, but I think it would come up more often. For example say that I take a flop with two other players in a ten handed game of limit Holdem. My hand is 2 cards of the same suit. The flop gives me a flush draw. The mathematics you can find in any book on poker theory now tells me that since I have nine outs with 2 cards to come I have approx 35% chance of drawing to my flush (assuming my opponents have none of my suit) It seems to me that this is very unlikely because 7 players were dealt cards that were folded preflop. Surely some of those cards that are now in the muck are my suit. In fact it would be a remote possibility that all of the cards either in the muck or in the hands of my opponents are not of my suit. So if I know that I must make my flush to win and I continue playing based on the fact that the pot is offering me correct odds for a 35% chance of winning could this be an error?

[lorinda]

If you knew your opponent had 77, you would fold your AK (unless you were getting short stacked and in danger of being blinded out) every time

Unless it was a ring game and the blinds tipped the EV in your favour at 46%

Lori

[Cyrus]

"Say that I take a flop with two other players in a ten handed game of limit Holdem. My hand is 2 cards of the same suit. The flop gives me a flush draw. The mathematics you can find in any book on poker theory now tells me that since I have nine outs with 2 cards to come I have approx 35% chance of drawing to my flush (assuming my opponents have none of my suit)."

I haven't seen any serious book addressing this kind of flop with the assumption that the opponents have none of the player's suit!

"It seems to me that this is very unlikely because 7 players were dealt cards that were folded preflop."

It's not simply unlikely, it is quite wrong to assume such a thing. There is simply no basis for that kind of assumption.

This is similar to the erroneous thinking of BJ players who assume things about the cards in the shoe, thinking they see clumps, sequences, or biases.

"Surely some of those cards that are now in the muck are my suit. In fact it would be a remote possibility that all of the cards either in the muck or in the hands of my opponents are not of my suit."

This is a fallacy. When you have no other (reasonably reliable) information about the cards in your opponents' hands, then the correct thinking is assuming that all the cards are out there and all the combinations are possible.
In other words, the cards of your suit could all be in the muck, they could all be in the deck waiting to be dealt or they could be in both places.

--Cyrus

____________________________________________

The mathematics tell us that when you hold 2 suited cards and the flop gives you a 4-Flush draw, you will make that Flush 35% of the time. And this is not on the basis of any assumption about what "the opponents are holding". It is based on the solid premise that there are still (13 -2 -2)= 9 cards of your suit that are out there somewhere, anywhere.

C(9,2)= 36 combinations that can be made, with 2 cards in each combination made out of the 9 remaining of your suit. This means that there are 36 combinations whereby you hit your suit in both Turn and River.

You can also hit 1 out only, in either the Turn or the River, and still make your Flush of course. This case will combine any one of the 9 remaining cards of your suit with one card from the rest of the cards, none of which is of your suit. The rest of the cards out there are (52 - 2 cards in your hand - 3 cards on the flop - 9 remaining cards of your suit) = 38, so you can have 9*38 = 342 combinations with 1 card of your suit and 1 blank.

36+342= 378 is the number of all the combinations for Turn and River that hit you with 1 or 2 of your outs for a Flush. If this is compared with the total number of all possible Turn+River combinations, which is C(47,2)= 1081, we get the probability of hitting either 1 or 2 outs and making your Flush. It's 378/1081 = 0.349676225716928769657724329324699 or 35%.

[Nottom]

</font><blockquote><font class="small">In reply to:</font><hr />
All the worring about there being a better chance of there being aces in the deck becasue people folded is pretty insignificant compared to the fact that you can't confidently bet 77 unless you spike your set.

[/ QUOTE ]

I think this is the most relevant portion of my post. Sure maybe based on a player betting habits you can adjust the odds slightly if you reads are accurate, but in a hypotheticl NL game if an opponent goes all in in front of you and shows you his 77 (assume the blinds are small enuf to not be a factor) short of another opponent flashing the other 2 7's, I think my AK is going in the muck.

[Cyrus]

"Say that I take a flop with two other players in a ten handed game of limit Holdem. My hand is 2 cards of the same suit. The flop gives me a flush draw. The mathematics you can find in any book on poker theory now tells me that since I have nine outs with 2 cards to come I have approx 35% chance of drawing to my flush (assuming my opponents have none of my suit).

It seems to me that this is very unlikely because 7 players were dealt cards that were folded preflop."

This is false but I guess one must elaborate a bit more than my blank dismissal.

Since we do not know (because if we do, it's a different ball game!) where the remaining cards of our suit are, we must assume that they can be anywhere. They could all be in the pack of cards waiting to be dealt, they could all be in the muck, some could be in our opponents' hands -- or they could be spread among the three places.

We compute mathematically the probability of hitting the Flush by assuming that the suited cards are somewhere among the 47 cards that are unknown to us : 52 - 2 cards in our hand - 3 cards on the flop = 47. Those 47 cards include the muck, the undealt cards, the cards still at play among our opponents, everything.

This may seem intuitively to be incorrect -- as you said, How can NO ONE of our opponents hold/mucked a single card of my suit??

It is nevertheless correct for the simple reason that the calculation takes into account ALL the possible cases, from the suited cards being out of play upto them being all in the deck still. The calculation essentially has "weighed" all the possible cases and since they are all equally probable (equaprobable), it correctly accepts that the cards can be anywhere among the 47 unknown cards.

If the other cards of your suit are all in the muck, your chance of hitting the Flush is 0%. (Same thing if the cards are all in your opponents' hands.) At the other extreme, if the suited cards are all in the deck and waiting to be dealt, the probability of hitting the Flush, if you go to the River, is no longer 35% but higher, actually 45% *. It should be obvious that the correct probability should lie somewhere between 0% and 45%. And since the cards waiting to be dealt (35 cards) are more than the cards in our opponents' hands (12 cards), it should "make sense" that the correct probability, i.e. 35%, lies much closer to 45% rather than 0%.

--Cyrus

_______________________________________________

* We take the original post's premise, with 7 players at the table. We assume that all the other cards from our suit are still in the deck. The unknown cards are now (52 - 2 cards in our hand - 3 cards on the flop - 12 cards in our opponents' hands) = 35. This makes for C(35,2)=595 possible combinations of Turn+River.

We still have C(9,2)=36 combinations whereby we hit a suited card in both Turn and River.

Then we have (35 - 9 cards of our suit) = 26 cards that can be combined with each of our 9 suited cards to produce a Turn+River whereby we hit only 1 of our outs -- and still make a Flush of course. That gives 9*26= 234 such combinations.

36+234= 270 total combinations whereby we make a Flush by hitting 1 or 2 outs, and 270/595 = 0.45378151260504201680672268907563 or 45%.

[Cyrus]

DRAWING TO A FLUSH WHEN YOU HAVE FLOPPED A 4-FLUSH DRAW AND ALL THE OTHER CARDS FROM YOUR SUIT ARE IN THE DECK :

<pre><font class="small">code:</font><hr>

Number of opponents Prob of completing
1 36.4%
2 37.9%
3 39.5%
4 41.3%
5 43.2%
6 45.4%
7 47.7%
8 50.3%
9 53.2%

</pre><hr>