Maths in Poker: Part 1
I’m James — hailing from Essex, a UCL grad, and working as a junior doctor in Hertfordshire. I’m an aspiring surgeon (not sure what type), love going to Japan, and also run a small company on the side (6med).
This year, I’ve set myself a target to get good at one exciting hobby — and one that perhaps might prop up my uncontrollable spending habits. It might seem unconventional, but I’ve decided to try and get good at playing poker.
A friend of mine recommended a book called ‘Show Your Work’ by Austin Kleon (highly recommend), which is almost self-explanatory from its title. In a nutshell, I want to share my process of learning poker, teach it back to you guys, and hopefully garner some fans/readers while doing so.
The first post will commence a small series on maths in poker. If you managed to make it through your primary school maths lessons, you should be fine. If not, then perhaps poker isn’t for you. This post will be relatively short, and a basic knowledge of poker is assumed. We’ll first talk about the ‘counting outs’ based on your hand at the flop and then jump into the concept of pot odds. Finishing off, we’ll touch upon the essential topic of expected value in poker.
Counting Outs
Before getting started, I want to lay out the basic ‘ranking’ of hand strength in poker just as a reminder (strongest to weakest):
1. Royal flush — A, K, Q, J, 10 and all the same suit
2. Straight flush — five cards in a sequence and all the same suit
3. Four of a kind — all four cards of the same rank
4. Full house — three of a kind with a pair
5. Flush — any five cards of the same suit, but not in sequence
6. Straight — five cards in a sequence, but not of the same suit
7. Three of a kind — three cards of the same rank
8. Two pair
9. Pair
10. High card (you have nothing except the highest ranked card)
The best way to illustrate what I mean by counting an ‘out’ is by using an example:
Say you’re holding J♥ and 2♥. The flop comes, and it’s 5♥, 4♣, 3♥. You have a pretty decent hand, and unless your opponent has something stronger, you’re on track to winning the hand.
There are 13 cards of each suit in the deck — you’re currently holding 2 of them, and another 2 are on the board. 4 of 13 hearts have been dealt, and so there are 9 hearts left in the deck. You need one of these cards to be dealt at the turn or the river in order to win. You, therefore, have 9 outs.
The first part of learning poker maths is gaining the ability to count outs quickly and accurately during live games. The flush draw example above was an easy one to figure out, but you have a total of thirteen types of hand combinations/draws that have differing numbers of outs. Here’s a handy table:
Pot odds / Odds of hitting your draw
Pot Odds
Now for the fun bit. When playing poker in cash games (basically where you’re able to throw in more cash if you run out), you should always be thinking about whether a ‘call’ makes sense mathematically.
First of all, you should consider the size of the pot and the bet you’re facing — in other words, the ‘pot odds’. If the pot is currently £12 on the flop and your opponent bets £3, this represents a pot odds of 12:3 or 4:1. You have to bet 20% of the pot to win it.
Odds of hitting your draw
Alongside calculating the pot odds, you should work the number of outs you have based on your hand and the flop. Once you’ve done that, you can work out your odds of hitting your draw. Using the example of the flush draw again — you have 9 outs, your chances of making that draw on either the turn or river is 36% or roughly 2:1. You can quickly work this out by multiplying the number of outs by 4 (9 x 4 = 36%).
What’s more useful is calculating your odds of hitting the next card from the flop (the turn): if you’re on a flush draw with 9 outs, you have an 18% chance or roughly 4:1 of hitting your draw on the turn. A shortcut, as you might have noticed, is to multiply the number of outs by 2 (9 x 2 = 18%).
Here’s another handy table with all the odds worked out for you (although you can just use the maths shortcuts mentioned):
Expected Value
Expected value (EV) is a very important concept in poker. If you were to go to Wikipedia, this would be its definition:
“In probability theory, the expected value of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
Thus, it represents the average amount one ‘expects’ as the outcome of the random trial when identical odds are repeated many times.”
In essence, it’s the amount of money you’re expecting to win (if + EV) or lose (if — EV) over time taking into consideration the size of your bet and the probability of you winning the hand.
We already have the tools to work out whether we have + EV. You simply pit your pot odds against your odds of hitting your draw (assuming it’s the best draw possible, i.e. nutted).
If your pot odds for a decision to call is 7:1 (£21:£3) and your odds of hitting your flush draw on the turn is 4:1 (9 outs), then you have positive expected value and should call (from a mathematical point of view).
If you make this call 5 times, the odds says you will hit your draw once on average. This means you stand to win £21 for every £15 you invest — not bad at all.
The point is to always call from a position of +ve EV, knowing that over time you will stand to make money rather than lose money (in theory).
I don’t doubt that it’s easy for amateur poker players to excessively focus on individual hands and sessions, but consistently making the right calls over time, even if on a losing streak, is absolutely key. There is already a considerable amount of variance in poker, and it makes sense that you should make sure the maths is working for you, not against you.
Conclusion
My journey with poker begins here — an eagerness to lose all my money but not before building a solid foundation in basic probability and poker maths. The first step is being able to calculate outs quickly and knowing the odds of drawing different hands in any given situation. I will then be able to work out my expected value by taking into account the pot odds.
There is so much more poker maths, and I’ll be sure to document my process of learning of all it as well as putting it into action later on. Aside from the maths, becoming good at poker encompasses techniques in emotional discipline, psychological warfare, reading body language, and strategy — all juicy topics for the future.
On a final note, I highly recommend a fantastic article on the concept of expected value, not just in poker but for life in general: https://fs.blog/2018/01/expected-value
“Poker is a skill game pretending to be a chance game.”
~ James Altucher
