GT-NO: Two ‘Toy’ Poker Math Logic Problems

When faced with an opponent whose hand you can narrow down a lot more than any solver could, your reading ability doesn’t do you as much good as it should if you can’t use your brain, as opposed to a computer, to deduce the best counter strategy to the range you have put him on.

Two simplified situations I asked my intermediate students about did not elicit the correct answers. It wasn’t too surprising given their lack of experience. But I am now wondering if many of the better players would inexcusably screw these questions up as well.

If you are a regular player who has trouble with them, I would advise getting off your computer for a while and instead study some simple math and poker books to reset your mind.

Question 1

It is a pure toy game. You and your opponent are each dealt one “card” that has a number (which can include a fraction, such as 76.32). You both ante one dollar. After looking at your card you can either bet the size of the pot ($2) or check.

If you bet, he can call or fold. If you check, he must check behind you. As in real poker the winner of the $6 pot is the one with the highest card if it goes bet-call, or the $2 pot if it goes check-check. If it goes bet-fold, you win the $2 pot.

He will only call you with 90 or higher. Thus, you will destroy him if you play many such hands and you bet every time. But hopefully you instantly see that betting every time is not the best strategy. You should check some hands. Which ones?

Question 2

You are playing heads-up no-limit hold’em with a $100 stack which your opponent matches. There are no antes or blinds. You have two queens. You somehow see that he has A-K offsuit. In spite of the fact that there is no initial money in the pot he bets X dollars (not knowing that you know his hand).

You know he will call if you move all in (and will move in himself if you raise smaller). You know that if you just call, he will check unless he hits an ace or king. You know that if he does hit that ace or king he will move all in. You know that if he misses the ace or king he will check and fold the flop unless he is getting about 3:1 odds or better. 

Your two options are to call or move in. Your correct play depends on the size of his original bet. Assuming Q-Q is about 56% preflop, is about 3% to hit a set while the A-K hits at least a pair, and that straight and flush chances are ignored, about how small must his opening bet be such that it is better to move in than to wait to see the flop?

Answer 1 

Obviously, you should bet your bad hands and your great ones. If you are dealt 17 and you check, you will pull in $2 17% of the time for an EV of 34 cents. If you bet it, you will win the $2 pot 90% and lose the $2 bet 10% for an EV of $1.60. If you are dealt 98 and you check you will win $2, 98% of the time for an EV of $1.96. If you bet, you will win $2 90%, win $4 8% and lose $2 2% for an EV of $2.08.

But what about 93? Even without doing the EV math, it should be obvious that it’s wrong to bet it. It makes you an extra two dollars when he has 90-93 but costs you two bucks when he has 93-100.

The rest of the time doesn’t matter. It should also be obvious that 87 is a bad bet. It will cost you a pot-sized bet 10% while stealing the pot only 3%. 78 steals 12% while losing 10%, so it’s a profitable “bluff” even though a check would usually win.

Hopefully if you didn’t get the answer before I wrote the above it is now apparent what the actual solution is.

You check 90-95 (because you are an underdog when called) and 80-90 (because it loses more pot-sized bets than the number of pots it steals.)

Answer 2 

If you move in preflop you are about 56%. Your EV is about $12. If you just call you will win almost the full 100 about 3% of the time. That gives you about three dollars in EV. You will win X dollars about 67% when no ace or king flops. That gives you an EV of .67×.  About 30% of the time you will lose X dollars (when he flops at least a pair, and his bet makes you fold.) Thus, your total EV when you just call preflop is approximately .67x -.3x + 3. It’s about a push when that equals $12.

3 + .37 x = 12
300 +.37 x = 1200
37x = 900x  = about 24

In other words, it’s better to wait and see what comes if his first bet is more than about $25.

For those who find this result surprisingly small, the main reason is that holding back money to get him to fold on the flop helps the queens who are facing six wins on the last two cards more than they help the A-K who is only facing two wins. On the other hand, if the first bet is very small you are better off putting him all in preflop unless you need to reduce volatility such as in a tournament situation. ♠

David Sklansky is the author of The Theory of Poker, as well as nearly two dozen other guides on gambling, poker, and other games. The three-time WSOP bracelet winner’s latest book, Small Stakes No-Limit Hold’em: Help Them Give You Their Money, is now available on Amazon. You can contact Sklansky at [email protected].