After he signed up for a few games, Fred sat next to me and asked what were the card odds against a runner-runner straight and what money odds did he need to chase.
"Suppose, holding Jh-10h the dealer flopped Qs-7d-2c. There would be three doubletons that would make you a straight: A-K, K-9, and 9-8.
"There are 16 ways to make each doubleton, so 48 doubletons would make you a straight. Since C(47,2) doubletons could fall on the turn and river, the probability of a backdoor straight is 0.0444, roughly equivalent to one out, or about 21.5-to-1 against.
"Suppose you play the hand 1,081 times with plenty of chips: seeing the turn would cost you 1,081 small bets. Eight times out of 47, the dealer would place a king or a nine on the turn for an open-ended draw, and 8 out of 47 times, the dealer would place an ace or eight on the turn for an inside draw, for a total of 368 times. Because the bets double on the turn, it would cost you 736 small bets to see the river card for a total investment of 1,816 small bets.
"Twelve times out of 46, the dealer would put a river card on the table that would fill your straight, for a total of 48 wins. Those 48 wins must at least return your investment, so we have this expression: 48 * (Pot - Drop + 7*b) > 1816*b, where 'b' means a small bet. The 7 small bets inside the parentheses are the bet you would make to see the turn, plus the bets you and one opponent would make to see the river card, plus the 2 you would win when you made your flush, assuming no raises. We solve the expression for the size of 'Pot' before the turn, finding that Pot > (23.5*b + Drop). If the drop is negligible at the stakes you play, then you must have pot odds of at least 22.5-to-1 to pay to see the turn card." "But I wouldn't do it that way," Fred protested, "If the turn gave me an open-ended draw, then I'd pay to see the river. I wouldn't stay with only a gut-shot." "In that case, your investment to see the river card would be 1,448 small bets and you'd make a straight 32 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, 32*(Pot - Drop + 7*b) > 1448*b. Solving the expression for the size of 'Pot' before the turn, we obtain Pot > (38.25*b + Drop). If the drop is negligible at the stakes you play, then you would have to have pot odds of at least 37.25-to-1 to pay to see the turn card."
"That's crazy!" said Fred, "Why would I need bigger odds to draw to an open-ended straight than I would to draw to both open-ended and gut shot? That can't be right!"
"Look at this way," I told him, "suppose you were playing $4-$8 hold 'em, there were five bets before and after the flop, and there were five players who saw all those bets. With $94 in the pot after the house rake and bad-beat drop (the small blind folded), you would have pot odds of better than 23.5-to-1 to pay to see the turn card. With those pot odds, you'd have a positive expectation even if you were drawing to an inside straight. Your decision to abandon the hand was wrong-headed.
"In other words, if you had big enough odds to pay to see the turn, then you'd also have money odds big enough to draw to the inside straight."
Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon & kokopellipress.com. E-mail your Hold 'Em questions to richardburke@comcast.net
