Calculating Hold’em Odds

	An Ace with a small kicker isn’t as good a hand as it might appear.

In our last tutorial we armed you with the key facts and figures that are the foundations of poker and how you can use maths to get to grips with every possibility that can unfold.

You might not think it’s that important to know statistics like there are 1,326 possible starting hands and 19,600 flops in Texas Hold’em, but if you don’t, you’re effectively making decisions blind and – worse still – not maximising your chances of winning.

One thing is certain: playing position and acting aggressively is a mathematically proven route to success. Raise before the flop and you might steal the blinds even if you’re not holding the best hand. If you get called you’ve got the momentum and should carry on betting because most hands miss most flops.

But in the spirit of getting you to understand the numbers, how can this be demonstrated? With a little help from that scientific calculator we advised you to buy last issue (one with a ‘combinations’ (c) button), you can easily talk yourself into getting more chips into the pot from the off (or decide that folding is actually a better option).

For example, say you raise with a random hand in late position, and your passive opponent calls with A-K in the big blind. What are the chances he will miss (and will therefore check and fold to your bet)? Assuming you have any two cards below A-K, like 7-3 off-suit or 5-5, the answer is 42 c 3 (the remaining combinations of three cards below A-K possible) divided by 48 c 3 (the total combinations possible), which is 66.37%. This means that whatever you’re holding, you’ll win the pot two-thirds of the time through aggression alone.

The Numbers Game

You can use the same theory to make calls on almost any situation that you might encounter playing poker. Say you’re in a tournament and you’re short-stacked. You’ve got less than ten big blinds left, so you decide that on the next decent hand you’re going to make an all-in move while you’ve still got enough chips to push people off the pot and make drawing hands pay to stay.

Here your decisions are determined pretty easily by a few calculations. Say you’ve got an Ace with a weak kicker – a Five for example. Now you might automatically assume that any Ace is a stick-on favourite, but is this correct?

Let’s start with you in the small blind when everyone’s folded round to you. You have 9500 chips and the blinds are 500-1000. Once you’ve put in your small blind of 500 you have 9000 left and can win 1500 by moving all-in – this means you’re getting odds of 1/6 on your money. Which isn’t great on the face of it. To make it a worthwhile move you should be looking at being at least that big a favourite to take the pot down.

There are 1225 potential hands that the big blind could have, but how many will he call with that are beating your A-5? This, of course, depends on many factors: stack sizes, what sort of player he is and so on. But for argument’s sake, let’s imagine the big blind is a tight player who will call with any pair Sevens and up, and A-8 through to A-K. There are six combinations for each pair (4 c 2) except A-A – there are only three (3 c 2) as you hold an Ace – and 12 combinations for each A-X hand (three Aces combined with four Kings, Queens, Jacks and so on). In total, then, there are 117 combinations beating you, or less than 10% of the hands he can be dealt, making moving in a profitable play.

What Are The Chances?

Now what if you were on the button with 9000 chips and two other players left? Well, assuming all the details are the same, each of the two players will call with 117 out of the remaining 1,225 (9.55%) combinations and fold the other 1,108 (90.45%) hands. And by multiplying together the chance of each of them folding (90.45 x 90.45 / 100) you get the chance that both will fold (81.8%). This means you’ll win the blinds about nine hands out of 11, which isn’t ideal when you’re only getting about 1/6 on your money. But you have to factor in the chance that if called you can double or even treble through holding the worst hand.

Underdog

A quick visit to an online odds calculator (like twodimes.net) reveals that against a bigger Ace you’re about a 3/1 dog and against pairs lower than Aces you’re about a 7/3 dog. For the sake of simplicity and because you’re more or less screwed if both players call or one finds Aces, let’s round these numbers up and say that when called you’re always a 3/1 dog. Now you can say that nine times out of 11 you will win the blinds (1500 chips) and two times out of 11 you will lose 75% of the time (a loss of 9000) and win 25% of the time (an average of 9750 depending on which stack calls). So when called your net result is 9750 x 0.25 – 9000 x 0.75 = -4312.25.

Now, to see whether this is a profitable play or not, the odds need to be factored in. Going on the information you have you’ll win 1500 nine times and lose 4312.25 twice. So the net result on the play each time you make it is 1500 x 9 – 4312.25 x 2 / 11, which is 443.22, making it a profitable play, but not by much. In fact you might want to think again before putting all your chips on the line. The moral of the story, then, is that an Ace with a small kicker isn’t as good a hand as it might appear for making your stand.

And now you understand the reasons why, you can apply the numbers to your overall game. Admittedly, we’ve used one very specific example here, but the general concepts can be used in any given situation. And now the winter nights are drawing in, what better way to while away the long, dark hours?

However, we don’t want to scare you off poker because of some scientific ponderings. As you know by now, poker is fun and we’re not suggesting you should turn it into an academic study. Play the game as you feel it and not strictly by the numbers. The maths just makes poker that bit more transparent, which if you can master, will give you the edge when you’re up against some tough players. And no one can dispute that poker is more fun when you’re winning.