Fred took me to task last Saturday at our local poker room. "You wrote that you count a backdoor flush draw as one out after the flop," he said, "because the odds are 23-to-1 against making a flush. David Sklansky wrote in an article for Poker Digest, Vol.3/No.9, that you need odds of 27-to-1 or larger to draw to a backdoor flush. What's up with that?"
"Mr. Sklansky is correct when you have a deep stack," I told Fred. "And he also pointed out that if you were to go all-in, then you would need pot odds of 23-to-1 for a runner-runner flush."
That stopped him cold. "How can the odds change depending on how many chips you have," he demanded, "that just doesn't make sense!"
"That's the paradox," I answered. "Suppose, holding 10h-9h, you went all-in after the dealer flopped Ah-4s-2d. Then C(10,2)/C(47,2) equals the probability of hearts appearing on the turn and river, about 0.0416, or 23-to-1 against.
"Now suppose you play the hand 1,081 times with beaucoup chips. Your pre-turn bets will cost you 1,081 small bets. 10 out of 47 times, the dealer will place a heart on the turn, for a total of 230 times. If there's a bet on the turn, then it'll cost you 460 small bets for a total investment of 1,541 small bets. Nine times out of 46, the dealer will place a heart on the river and so you'll make a flush 45 times. To gain a profit you must make up for all those times you missed, plus pay the house rake, bad-beat, and dealer toke. This expression applies, 45*(Pot - Drop + 7*b) > 1541*b, where 'b' means a small bet. The 7 small bets inside the parentheses are the bet you made to see the turn, plus the bets you and one opponent would pay to see the river card, plus the 2 you would win when you make your flush, assuming no raises.
"We solve the expression for the size of 'Pot' before the turn, finding that Pot > (27.2*b + Drop). If the drop is negligible at the stakes you play, then you must have pot odds of at least 26.2-to-1 to pay to see the turn card.
"That's the answer to the paradox," I told Fred, "your cards odds don't change, the pot odds that you need to draw to the flush change because of those bets on the turn.
"Actually, you probably need odds larger than 26.2-to-1 because you don't want the board to pair. With multiple opponents and multiple raises before and after the flop, your opponents probably have a set or two pairs, so you surely don't want the board to pair, which rules out the 4h and 2h. The expression then becomes 28*(Pot - Drop + 7*b) > 1449*b, which simplifies to Pot > (44.75*b + Drop). So you'd need pot odds of about 44-to-1 to proceed with even the nut flush draw. With less than the nut flush, you'd need even larger pot odds."
"Then why," Fred asked, "do you count a backdoor flush draw as one out. It's rarely worth 4 percent: it's worth about 2 percent according to your last calculation."
"Don't you agree that a backdoor flush draw increases the value of your hand?" (He did.) "If you have anything else going for your hand, say over-cards or a pair, then counting one out for a backdoor flush draw might sway a close decision. Once in a while the dealer will lay out runner-runner trumps and you'll rake a nice pot that otherwise would go to the enemy."
Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon.com & kokopellipress.com. E-mail your Hold 'Em questions to
richardburke@comcast.net
