The complex mathematics of poker: simplified.
There's some incredibly complicated maths involved in poker games. Unlike fixed odds games where probability can be assigned clear, discrete values, the skill element in poker makes this tricky. Poker odds are best defined according to three variables: expected value (EV), pot odds and implied odds. This page will provide a basic overview of these variables and explain how they affect your chances of success in poker.
Expected value (EV) is the amount of money you can expect to win or lose on average by making a certain play, calculated by adding together the probability of every possible outcome and multiplied by the payoff. The resulting figure is the amount one can ‘expect’ from a given wager, all things being equal.
Put simply, EV is the average amount you expect to win or lose on a bet. If you expect to make a profit, a bet is described as +EV. If you expect to lose, it is –EV.
Let’s consider an example in poker (courtesy of pokerforum.com):
You have AH, JC in your hand. It comes to the river and AC, TC, 5D, 8C and 3C is on the board. You are in first position, with a £100 pot and the big bet is £10.
First, we can reasonably assume the following:
- Your opponent will fold if they do not have a club.
- They will raise with KC.
- If you check, they will bet with a club and check with no clubs.
There are 8 clubs in play, any of which could be in your opponent’s hand, 6 of which will lose to your hand. Therefore, the chance of your opponent calling is 6/8:
£10 * 6/8 = £7.5
If your opponent raises, implying the King or Queen, you will lose £10:
-£10 * 2/8 = -£2.5
Therefore, the EV of betting is £7.5 + -£2.5 = £5. Thus, betting would be a +EV play, but can you do better?
If you check, with the intention of raising if you opponent bets, then the numbers start to look a little different. Assuming, again, that your opponent will raise with the nuts and only bet on a club, you will lose £20 if they have KC or QC:
-£20 * 2/8 = -£5
If your opponent has any other club, you win £20:
£20 * 6/8 = £10.8
Thus, this scenario represents the play with the highest +EV.
The problem with playing according to EV is that it doesn’t really work if: a) you’re up against a strictly psychological player who deliberately plays against the odds b) you’re playing a novice who doesn’t understand the odds.
Pot odds are actually one means of determining the expected value of a poker action. Pot odds are the ratio of the current size of the pot compared the size of a bet. For instance, if the pot stands at £30 and the cost of a call is £10, then the pot odds simplify to 3:1 (equivalent to ¼ or 25%.)
Pot odds can be used to determine the EV of a drawing hand (a hand that is currently worthless, but has potential if you hit the right cards on the board.)
In this case, the EV of a call is determined by comparing the pot odds to the odds of drawing a winning card. When the odds of drawing the right card exceed the pot odds, this play is +EV, if not it is –EV.
Implied pot odds
Implied odds indicate how much you’re expected to win if you make your draw: there is no mathematical formula for working them out.
If you think you can extract more money from your opponent when you make your draw, then you have good implied odds and should call. If you make your draw but don’t think you can extract more money from your opponent, then you have bad implied odds and should fold.
A lot of time in poker you will not be getting the right odds to call, but you know that if you do call and make your hand, you will be able to get more money from your opponent. In these situations it’s fine to call with bad pot odds if you think you can win a greater amount later in the hand.
Reverse implied pot odds
Reverse implied odds are the opposite of implied odds, in that you can estimate how much money you will lose after you complete your draw.
Take the following example: you have 5C 7D and the flop is 8H, 9H, TD and three players have seen the flop. You can make a straight with any J and any 6. However any J or 6 that is a heart could give another player the flush and any player who holds a Q will make a higher straight if the J lands.
It would be better to fold this hand on the flop if any further action took place, as it would be highly likely that if you did make your hand, another player would make a better hand.
Regardless of the poker game you’re playing, you’ll be dealing with variance whether you like it or not. Variance is the up-swings (winning money) and down-swings (losing money) that you get with playing any form of poker.
To give you an example, pocket AA versus any random hand is a 85.2% favourite to win the showdown. If you have pocket AA pre-flop and you manage to go all-in, you’re expected to win approximately 4 times out of 5. However, in poker you can go all-in five times and lose all five (less than 0.1%).
On the flipside, you could have a random hand and be up against AA, and win all five times.
This is what we mean by variance in poker, although you’re statistically likely to win, you don’t. The key with variance is that over an infinite number of hands it will all even out back to neutral. Even though in your last lesson you lost pre-flop with AA five times in a row, over the course of a week, month, or year, your variance with eventually even out.
Most new poker players fail to learn this concept and claim that poker is 'rigged' or they are just extremely unlucky. Even the world’s best players face variance, and it’s not uncommon for them to lose as many as 20 buy-ins in a single day, only to recoup their losses later on in the week or month.
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