I'm posting one puzzle, riddle, math, or statistical problem a day. Try to answer each one and post your answers in the comments section. I'll post the answer the next day. Even if you have the same answer as someone else, feel free to put up your answer, too!
Thursday, June 10, 2010
Long Odds on Drawing an Inside Straight
It's often said trying to draw to an inside straight is a fool's bet. Why is that?
For an inside straight, your chances are 4 out of 52. If you're going for an inside straight flush, then it's one out of 52. I don't go for inside straights but the few times I have, it never came in.
Let's say you're playing 5-card draw--that will make this comparatively simple.
If you're drawing to an inside straight, it means that you have four communally non-consecutive members of a series of five consecutive cards (e.g. 4,5,6,8), and one other card not matching this pattern. Out of the full deck of 52 cards, you've now seen 5--leaving 47 cards unseen. Some of these cards will be in other players' hands--but, since the chances of any given card being in their hands vs. still dealable are equal, this is not important.
We're dropping one card and drawing one replacement from the 47 possible remaining cards in the deck. There is one value (e.g. 7 in the above example)--and that value is still available in all four suits (since none of the cards we still hold have this value, nor does the card we've discarded). This leaves you with 4 chances for a successful inside straight draw out of a possible 47; so your odds of success are 4/47 = 0.085106. For even money, or anything close to it, this is a terrible bet; if you run this situation out many times, you'll win only 4/47ths of the time--so you'll lose lots and lots of money.
Now, dropping one card for the inside straight draw also gives you a shot at drawing a pair. You hold four cards that have possible pairs left to be drawn, and (assuming you didn't have a pair before your discard) each has three possible matches still dealable. Your odds of making one of these are 12/47 = 0.255319--just a bit better than one in four. If you think a pair is good in the hand (depending on the actual values of the cards you hold--high-end or low-end), then you actually have 16 outs out of 47 if you choose to drop the one card and "draw for the straight". That's 16/47 = 0.340425; so you'll make a hand fully 1/3 of the time. If your card are suited, your odds obviously improve even more; subtracting the outs we've already named (and assuming you don't already have a flush before the draw), there are 13 - 1 (straight card--actually straight-flush card) - 4 (cards in your hand) = 8 cards that will make your flush (independently of any of the above hands) if your four cards are suited. This increases you to 24 out out of 47 to make a hand--24/47, or 0.51, which is not bad at all. If you've got suited cards and an inside straight draw, you're fairly certain that a pair is sufficient to win, and you're getting even just even money on the bet, you should go for it.
Of course, the bet is the important part in the decision. Usually a pair isn't good in this situation, and usually you're not also drawing to a flush, so let's go with the straight-up inside straight draw odds of 4/47. If the ratio of the call you have to put in to the pot at stake is good enough, you'd still be obliged to call. For instance: Suppose you're drawing twice, rather than once; and that, in the first round, the pot was raised up to 50,000. In the second round, after your first draw, you're left with a possible inside straight draw--but your opponent, sensing you have nothing solid, bets an additional 1,000 to move you out of the pot. If you make the inside straight draw in this scenario many times, 4/47th of the time you'll win, and you'll make a gain of 49,000 each time; and 43/47th of the time you'll lose, losing 1,000 each time. Your average yield on the decision, then, is (49,000 * 4 - 43,000)/50 = 3,060; so the bet yields a net gain, and you should decidedly take it.
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Because there is only one card out of 52 that will satisfy your draw.
ReplyDeleteEl wrongo, betting breath. You have 4 cards out of 52.
DeleteTrying to draw an outside straight will give you twice the chance to get the right card. But that's pretty relative, isn't it?
ReplyDeleteFor an inside straight, your chances are 4 out of 52. If you're going for an inside straight flush, then it's one out of 52. I don't go for inside straights but the few times I have, it never came in.
ReplyDeleteerr... If you have at least 4 cards in your hand already... plus the card you may have thrown back?
ReplyDeleteLet's say you're playing 5-card draw--that will make this comparatively simple.
ReplyDeleteIf you're drawing to an inside straight, it means that you have four communally non-consecutive members of a series of five consecutive cards (e.g. 4,5,6,8), and one other card not matching this pattern. Out of the full deck of 52 cards, you've now seen 5--leaving 47 cards unseen. Some of these cards will be in other players' hands--but, since the chances of any given card being in their hands vs. still dealable are equal, this is not important.
We're dropping one card and drawing one replacement from the 47 possible remaining cards in the deck. There is one value (e.g. 7 in the above example)--and that value is still available in all four suits (since none of the cards we still hold have this value, nor does the card we've discarded). This leaves you with 4 chances for a successful inside straight draw out of a possible 47; so your odds of success are 4/47 = 0.085106. For even money, or anything close to it, this is a terrible bet; if you run this situation out many times, you'll win only 4/47ths of the time--so you'll lose lots and lots of money.
Now, dropping one card for the inside straight draw also gives you a shot at drawing a pair. You hold four cards that have possible pairs left to be drawn, and (assuming you didn't have a pair before your discard) each has three possible matches still dealable. Your odds of making one of these are 12/47 = 0.255319--just a bit better than one in four. If you think a pair is good in the hand (depending on the actual values of the cards you hold--high-end or low-end), then you actually have 16 outs out of 47 if you choose to drop the one card and "draw for the straight". That's 16/47 = 0.340425; so you'll make a hand fully 1/3 of the time. If your card are suited, your odds obviously improve even more; subtracting the outs we've already named (and assuming you don't already have a flush before the draw), there are 13 - 1 (straight card--actually straight-flush card) - 4 (cards in your hand) = 8 cards that will make your flush (independently of any of the above hands) if your four cards are suited. This increases you to 24 out out of 47 to make a hand--24/47, or 0.51, which is not bad at all. If you've got suited cards and an inside straight draw, you're fairly certain that a pair is sufficient to win, and you're getting even just even money on the bet, you should go for it.
Of course, the bet is the important part in the decision. Usually a pair isn't good in this situation, and usually you're not also drawing to a flush, so let's go with the straight-up inside straight draw odds of 4/47. If the ratio of the call you have to put in to the pot at stake is good enough, you'd still be obliged to call. For instance: Suppose you're drawing twice, rather than once; and that, in the first round, the pot was raised up to 50,000. In the second round, after your first draw, you're left with a possible inside straight draw--but your opponent, sensing you have nothing solid, bets an additional 1,000 to move you out of the pot. If you make the inside straight draw in this scenario many times, 4/47th of the time you'll win, and you'll make a gain of 49,000 each time; and 43/47th of the time you'll lose, losing 1,000 each time. Your average yield on the decision, then, is (49,000 * 4 - 43,000)/50 = 3,060; so the bet yields a net gain, and you should decidedly take it.
too much time on your hands
ReplyDelete