Note: Not at the old Poker1 site. A version of this entry was first published in Card Player. This entry in the “Aunt Sophie” series covers poker.
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Michael Wiesenberg
Aunt Sophie draws to an inside straight (Part 1)
“Can you believe,” chuckled Aunt Sophie, as we strolled about the La Brea tar pits, “a player in the draw game actually came in to draw to an inside straight? That must be the worst play in draw poker, right?”
“Well, Aunt Sophie,” I temporized, “there are a lot of bad plays in poker, but, as you know, some bad plays are not always bad.” I carefully stepped over a small stream of black tar welling out of a hole in the middle of a gently sloping meadow. “In fact, there are times when a play that is bad most of the time is actually good.”
“So?” she demanded. “Tell me when drawing to an inside straight is good.”
“I will,” I responded, as we leaned on a railing overlooking a pond in which sheets of tar floated. “But first let me reply to your question about drawing to an inside straight being the worst play in draw poker.”
Assume
I watched gassy bubbles float lazily to the surface, where they exploded darkly. Just below us, a prehistoric ancestor of a hippopotamus tried to fight its way out of the entrapping tar into which it had wandered after a sip of water. While that beast was merely a cement fabrication, beasts like it had done the same thing, and their bones were preserved through the centuries to be dug up by archeologists at the start of this century. “Let’s assume,” I continued, “for the sake of argument, that drawing to an inside straight is always bad. You make a straight only once in about twelve tries, so it’s about eleven to one against you. If you draw to a small pair, you improve that hand approximately one out of four times, or three to one against. That would seem to make the draw to an inside straight a much worse proposition than the draw to a small pair.”
A small boy picked up a large rock, about to toss it over the chain link fence into the tarry pond. A man in a beige uniform gently tapped him on the shoulder, and the boy laid his burden down.
“But, consider,” I went on, “that you get an inside straight draw about once in 7.8 hands, while you get a small pair maybe once in 2.6 hands. By ‘small’ I mean jacks or less. Anything less than openers most of the time. I include jacks, because a lot of times players call with jacks when someone has already opened. Generally you should play jacks in a limit draw poker game only when you can open with them, and that only in late position. So, you get a small pair nearly three times as often as an inside straight. That means that even though the hand improves more often, it shows up more often. You already know that playing a small pair is a bad play for two reasons. One is that when someone opens, he or she has your small pair beat. If neither of you improves, you don’t win in a showdown. Of course, you can say that the player who draws to the small pair can bluff after the draw, and that’s true. But the opener can call that bet often enough to make it not a profitable play. Using game theory to call the bet should make it an absolutely even play in the long run if the bettor also uses game theory to make the bet. Usually such a player does not play according to game theory, however, and the caller can easily adjust her calling strategy. If the bettor bets only when improving such a draw, the caller folds most of the time. If the bettor likes to bluff, the caller calls more often. Pretty obvious.”
Aunt Sophie settled onto the bench facing the pit. We could see beyond the grass and trees and tarry ponds the incongruity of buildings and LA traffic. “Mmm hmm,” she mused. “The bet after the draw either breaks even or worse than even for the player who draws to the short pair. And the before-the-draw bet, that is, calling to draw to a small pair, loses overall.”
Opener improves more often
“That’s right,” I agreed. “And of course, it’s worse than that. Most of the time, neither player improves after the draw. The opener actually improves more often than the caller, because the opener has aces a certain percentage of the time, and, in a three-card draw, aces improve the most of any pair because there are five of them in the deck. Also, drawing to ace-joker a player can end up with a straight or flush, which can’t happened with any other draw. But of course the worst to happen is when both players improve equally. They both make two pair, or both make three of a kind. Then there’s a bet after the draw, and the short pair usually loses that. Oh, yeah, sometimes the short pair improves more than the opener. He might make a full house when the opener makes three aces. Then the three aces loses several bets. But that balances out, and, again, slightly in favor of the opener. Sometimes both players improve, and the opener improves more than the caller. Then the caller loses several bets after the draw.”
I closed my eyes, imagining what these meadows would have looked like a thousand years ago. Grasslands, very dry. Animals searching everywhere for open drinking water, happening upon these treacherous tarry ponds. The remains of one woman had even been found here, displayed in the museum up the hill from us. Speculation was that she had been thrown in by other members of her aboriginal tribe for various transgressions. “Now, what happens in drawing to an inside straight. Either you make it, and it wins. You usually pick up an extra bet after the draw if it does. More often, you don’t make it. But all you lose is the one bet before the draw. You either improve a lot, or not at all. You don’t have the cases that are so bad for the player drawing to short pairs of improving just enough to lose a bet after the draw, or a lot, and losing several bets to another player who improved equally. Yeah, the player will make his straight and still lose once in a while, but not very often. And he’ll beat three of a kind more often than he loses to a better hand, so that more than offsets the few losses. So, the inside straight draw loses one bet most of the time. Since the draw comes up a third as often as the small-pair draw, it loses that bet only a third as often. The pair draw loses one bet part of the time, and several bets part of the time. And it does both three times as often. So, the conclusion to be drawn from this?”
“That drawing to a small pair,” answered Aunt Sophie, “even though it improves more often, is a much worse play than drawing to an inside straight. Okay, I agree. But you said that it isn’t always a bad play to draw to an inside straight. That sometimes it’s a good play. How can that be?”
“That, my dear,” I replied, “I’ll tell you next time. I can see a meter maid approaching, and our two hours are up.”