Mathematics of Poker
Poker Mathematics can seem daunting to the beginning player – but is actually quite simple. Here we look at the key concepts in using poker mathematics to make better decisions while playing – and thus to increase your profits.
The concept behind all of poker mathematics is ‘expectation', that is whether a bet – if replayed 100's of times – will win or lose money over time. In poker the idea is that if your bets, raises and calls have a positive expectation you will make money over time, regardless of the outcome of the current hand. Conversely it is possible to make a ‘negative expectation' move, get lucky and win a big pot. However making negative expectation bets or calls will lose money in the long run.
The question thus becomes – ‘how do we know if a bet has a positive or a negative expectation?'
‘Pot-Odds' and ‘Outs'
There are two basic principals to learn here, they are ‘Pot-Odds' and ‘Outs'. Pot-odds are simply a way of assessing the price, compared to the money that is already in the pot, that you are paying to see the next card. Outs refer to the number of unseen cards in the deck that will help your hand – compared to the number that will not. We will explain both concepts further with the help of an example:
You are dealt A-4 of spades and call a small bet before the flop in a Texas Holdem game. The flop comes down 2-8-Q with 2 spades showing, the pot is $5. Your opponent bets $1 and everyone else folds – time to look at your pot-odds and your outs.
Here you have to call $1 into a pot totaling $6 – your pot-odds are exactly 6/1. We can then compare these odds with your chances of improving to a flush on the turn. Out of a deck of 52 cards we have seen 5 – the 2 in your hand and 3 from the board. This leaves 47 cards, 9 of which are spades (we have already seen 4 of the 13 spades in the deck). This means that the chances of the next card being a spade are 9/47 – a little over 4.2/1.
So, the pot is offering 6/1 and our chances of improving are 4.2/1 – if we ran this bet 100's of times we would show a positive expectation, clearly we have a profitable call.
Pot-odds can be used for all scenarios where we think we need to improve to win the hand. They can also be used when comparing our chances of winning to the size of the pot. Imagine you reach the river holding J-J on a board of 2-3-7-10-K. Your opponent makes a small bet, $1 into a pot of $9. Here we are getting 1/10 pot odds, the decision to make is whether we believe – based on the history of this opponent and the betting so far – that we will win the pot more than 10% of the time. If you believe that there is a 50% chance that this opponent has a pair of kings a 40% chance a pair lower than jacks and a 10% chance he is bluffing then we have a clear call. We are 1/1 to win and getting 10/1 odds, a clear positive expectation call.
Pot-Odds alone do not account for those times where you will win more money on later betting rounds if you make your hand. For example you have 6-7 on a flop of 5-8-A, your opponent bets and you calculate your pot-odds as being 3/1 with chances of improving to the 8 outs needed to make a straight as 4.5/1. This looks like a negative expectation bet. However, if you take into account that you may win 2, 3 or even 5 bets on the turn and river betting rounds, then a call may be in order.
These later bets are known as your ‘implied-odds'. When calculating your odds and outs an estimate of what you might win on later streets can be very useful. In the above example if your opponent bet $1 into a $2 pot you would have 3/1, if you expected to win $5 more those times that you do make your straight then your true odds are 8/1 – you have turned a negative expectation call into a positive expectation one.
Beginning poker players should be cautious when estimating implied odds. Many factors affect this including the tendencies of specific opponents, how obvious your draw is and your own table image. Being cautious in your estimate will ensure that your situation is genuinely profitable. Knowing the basics of poker mathematics will improve your game when playing with any free poker cash offer. Do not forget to read our article about Poker Psychology.
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