Math is hard

Lord knows I've made more than any blogger's share of mathematical mistakes in the course of five years of writing about poker. I kind of doubt, however, that I've ever made as many in one post as my pal Very Josie did earlier today.

Here's her first example:

I’m holding a KQ of spades. The flop comes ace of spades, 10 of hearts and 2 of spades. I have a nut flush draw and an inside straight draw. What are the odds that I hit one of these great hands? Hmmm…First thing to do is to count how many cards are out there that will complete my hand. 4 spades are showing, so there are 9 left that will give me a flush; 9. There are 4 jacks in this deck that will give me a straight; 4. Nine plus four is thirteen. Surely you don’t need to be an accountant to figure that out.

It appears that being an accountant is not enough to get it figured out correctly. The jack of spades is being counted twice here--once as a spade and once as a jack. The number of cards that will complete either a flush or a straight is 12, not 13 (as Josie acknowledged after a commenter pointed this out).

There are 13 cards that will give me a big and most likely winning hand. To determine the odds of hitting one of these cards on the turn you take your 13 outs and multiply that by four. 13 times 4 is 52. I have a 52% chance of hitting my winning hand on the turn. If I do not get my card on the turn, it’s time to calculate the odds of hitting it on the river. You take those same 13 outs and this time multiply them by 2. 13 times 2 is 26. I have a 26% chance of hitting my hand on the river.

No, no, no. The "rule of four" gives you the approximate probability that you will hit one of these cards on either the turn or the river. Look at it this way: Each card constitutes about 2% of the deck, so the probability that the next card the dealer shows is, say, the eight of hearts is about 2%. If there are 12 favorable cards whose position is unknown to us (i.e., they might be in the stub of the deck in the dealer's hand, might have been dealt to an opponent, or might be among the burn cards), there is a roughly 2 x 12 = 24% chance that one of them will appear as the turn card on the board. There is another roughly 24% chance that one of them will appear on the river. Combined, that yields about a 48% chance of hitting one on either the turn or river.

A commenter calling himself "Four Hands" correctly made this same point on Josie's blog:

the Multiply by 4 is to calculate the odds of hitting on the turn _OR_ the river. It important for calculating whether or not to go all-in on the flop, but it not accurate if you're calling for a single card, or comparing pot-odds unless you're going to be all-in.

The odds of hitting on the turn are the _same_ as hitting on the river, well, slightly different because you've seen one card, but close enough that the approximation is usually fine.

To which Josie responding, puzzlingly:

NO 4 HANDS! I think you're saying the odds between the turn card and river are pretty much the same, except for that one measly card. i disagree because after the flop you have two chances to hit your hand, yet after the turn you have 1 chance, which is 50% less chance of hitting your hand.

see what i'm saying?

Josie has confused here two different quantities. One is the probability of hitting one of the desired cards on the turn. The other is the combined probability of doing so on either the turn or river. They are both potentially useful numbers, but they are completely different. If your decision is simply whether to call a bet on the flop in order to see the turn card, then the 2 multiplier is your approximation, because you don't yet know if your opponent will bet again on the turn, nor how much he might bet. On the other hand, if one or both of you will be all in on the flop, then you're getting both the turn and river, and you'll be interested in the combined probability that a desired card will hit either spot. "Four Hands" had it exactly right.

Josie's post said that she is twice as likely to hit one of her outs on the turn as on the river. A moment's reflection should reveal that that is an absurd conclusion.

Her next example:

I have J-J, which is definitely okay. The flop is 4-4-8 rainbow (all different suits). I have an over pair and I’ll come out betting here. The question is, what are the odds of improving my great hand. Any jack or four will give me a full house, and there are two jacks and two fours left. I have 4 outs. Four times four is 16. I have a 16% chance of hitting a full house on the turn and since four times two is eight, I have an 8% chance of hitting that full house on the river.

The problem here is more conceptual than mathematical. When I'm in a situation like this, the probability of improving my two pair to a full house is the last thing on my mind. The far more important question is whether I have a better hand than my opponent right now. If he called me pre-flop with Q-Q or K-K, I'm in deep doo-doo--so deep that even making a full house by another 4 hitting the board won't get me out of it. On the other hand, if he called me with 10-10 or 9-9, he's the one that's deep in the doo-doo. The probability of me catching another jack is so low that it's not worth basing any decisions on. It's true that a 4 improves my hand, but it improves the other guy's hand equally, so it's not really a meaningful consideration. If he already has a 4 or 8-8, I'm toast, and only a jack will change that, not another 4.

Another problem with the language here is the use of the term "out." Outs are defined as cards that will improve a currently losing hand to a winning one. If you're already ahead, it's nonsensical to count your outs, or even to speak of having them; it's the other guy that has to be looking for outs. So when Josie says that she has "4 outs," it means that she either thinks she's behind here or doesn't understand the whole concept of outs. And, again, even if it's the former, the number of outs is actually just two, because the remaining two 4s don't move her from being behind to being ahead. Only the jacks can do that. Having a full house is still a losing proposition if the other guy has a bigger one.

But fundamentally this situation does not pose a mathematical problem. It's a hand-reading problem. I have to figure out whether I'm winning or losing. Most of the time it's a favorable flop for me; I was ahead with the jacks before the flop and remain so after the flop. But once in a while an opponent will have smooth-called pre with a bigger pair, or got very lucky with either 8-8 or a 4 in his hand. My primary tasks are (1) to extract the most value from my opponent if he has a pocket pair smaller than mine (which is just about the only thing he could have with which he might pay me off), or (2) spend as little as possible to determine that my hand is second-best on those occasions that he has 8-8 or a 4.

Josie's final example:

We’re playing with the two and three of hearts. The flop is 4 of spades, 5 of hearts and Q of clubs. Right now I have an open ended straight draw and there are 8 cards in the deck that will give me a straight. After the flop I take the number of cards out there that will help me (8) and multiply that by 4 to get 32. There’s a 32% chance I will hit my straight on the turn.

This is wrong, for the reasons given in the first example. It's closer to 16%. To be exact, there are 47 unseen cards, and 8 of them make the straight, so it's 8/47, or 0.170, or 17.0%.

Alas, the turn is a king of hearts. In addition to my open ended straight draw, I also have a flush draw. Now there are 15 cards in the deck that I want. If one of them hits on the river, I’ll have either a straight or a flush. 15x2=30. I now have a 30% chance to hit one of my hands.

That's about right. It usually works out to actually be about 2% more than the rule of thumb predicts. Here, for example, there are now 46 unseen cards. 15/46 = 0.326, or 32.6%. But the difference between the quick estimate of 30% and the actual value of 32.6% will essentially never matter to a poker decision. Either way, you're next going to translate it into odds (about 2:1) for purposes of determining whether a call is worthwhile.