# Poker Mathematics

In addition to tells, intuition and feelings a very important part of what makes poker more a game of skill than a game of chance is the mathematics of the game. In this article we’ll introduce you to a few common terms in the poker game and then a few that aren’t so common. We’ll explain some mathematical concepts, and hopefully, with a little bit of luck, change the way you think of the game of poker.

### Poker Math for the Beginning Poker Player

We’ll start out with some terms that are pretty easy to visualize, and you may already have heard of or have a pretty good understanding of then we’ll work our way into more difficult topics from there.

## Pot Odds

The concept of pot odds is something that’s taken into consideration when you do not have the absolute nuts. This can happen in two situations, well three.

- You have a really good draw. Perhaps you flopped an open ended straight-flush draw. That’s a pretty good draw, the 2
^{nd}best in fact. You don’t have the nuts but, you sure do have a shot at winning the pot. Usually in this case, your odds of winning the hand are actually better than your opponents. - Reverse the above, you’ve made a hand, two pair perhaps but, you’re pretty sure you’re opponents working with a decent draw.
- Maybe you just want to take a stab at a nice looking pot with nothing at all, see if you can knock your opponent off of his hand; this is especially effective if he hasn’t made his hand yet.

Using pot odds, you will decide the amount of equity, and possible future ROI that you have in the pot, decide if you should or should not call, fold or even raise into a pot. This decision will be made based on the size of the bet you’re facing or are considering making in relation to the size of the pot and your odds of winning.

Sounds pretty convoluted doesn’t it? It’s really not. Let’s break it down and see what’s involved.

The simplest way to look at Pot Odds only involves a little addition and a quick bit of multiplication. If you’re one of those people that mentally shuts down when math is brought up… stick with me here, we can do this.

You’re holding 89 of hearts, the ten and jack of hearts are both on the board, as is the ace of spades.

- Count your outs.
- Multiply them by 4.

The answer to that equation is the percentage chance that you make your hand by the river. So, in the scenario above we have 15 outs for making a straight, a flush or even a straight flush. 15 x 4 = 60 so there’s a 60% chance that you’ll make a straight or flush by the river.

This is called the rule of 4.

If you have a lesser hand say you have Ace Ten and your opponent has obviously hit his king making a pair of kings on the flop. You have 3 outs. 3 times 4 is 12 so you have a 12 percent chance of hitting that ace by the river.

### What are the odds of hitting a flush after flopping a Flush Draw?

If you flop a flush draw you’ve got 9 outs. Using the rule of 4 we multiply 9 x 4 which gives us 36. So you have a 36% chance of hitting the flush by the river.

Let’s go back to scenario number 1. You’re on a straight flush draw, with 15 outs, and a 60% chance of making the flush by the flop. There’s $100 in the pot and your opponent bets $20. $20 is 20% of the pot, which makes for an easy call, even an easy raise.

This is where pot odds come in. If the % of the pot is = to or less than the % of the bet you have pot odds to call. So, as long as your opponent is betting less than 60% of the pot, you’ve got odds to stay in that pot.

The same with the flush draw from earlier, there was a 34% chance of hitting a flush by the river so, the most you’d want to invest is 34%.

If your opponent made a pot sized bet here, or even bet 50% of the pot, then pot odds imply it’s time to fold gracefully, or ungracefully, whichever suits you.

## Implied Odds

Earlier we talked about Pot Odds, using concrete numbers. The amount of money that is currently in the pot is considered so that a player can make a more logical decision when it comes to when to ‘chase’ and when to muck.

Implied odds take money into consideration that isn’t in the pot just yet, but is expected very soon. As a general rule nearly any bet can be called as long as you’re getting 3.5 to 1 on your money. That’s why they say you can limp with crap as long as you’re the 4^{th} person to do so.

## Pot Odds Vs. Implied Odds

When we talked about Pot Odds we looked at a scenario where there was $100 in the pot, and the bet was $20, or 20% of the pot. If you did happen to hit your draw however, it’s likely that you could get your opponent to either bet into you again, or to call a small bet, even a tiny equity bet. So you won’t just win that $100 pot, you’ll win whatever you get out of your opponent, which raises your implied odds, and gives you a good excuse to call a lot more bets!

Let’s look at an example. You’re holding a pair of 5’s. 1 player has called the $6 bet in front of you, and now it’s your turn you also limp in for $6. Counting the big blind, and the 1 caller before you there you’re getting 2 to 1 on your money. Counting the small blind, you’re getting 2.5 on your money, so you need one more caller, and you’d have that 3.5, and you could justify getting to a flop with your pocket 5’s. You’re in terrible position here by the way, so this might not be the smartest plan you’re cooking up, but we’re talking about math here, not position, power and patience.

You’re fairly sure that the player a couple seats down will call, she always calls, and her neighbor plays pretty loose too. As long as one of those two players call, you will have odds, what you have right now though, are implied odds.

Right now you have 2 outs though, and you need to hit a set to improve this hand. The odds of that happening are actually about 7.5 but, if we’re going to stick with the rule of 4 that we learned in our article on pot odds, we’ll multiply the 2 outs by 4, which gives us an 8% chance of hitting trips on the flop. That means we’d need at least 7.5 to 1 on our money. Imagine the money you’ll fail to leak using this strategy.

### The money you WON’T save using this mathematical poker strategy

Now I could be wrong here but, in the general flow of things, players learn to play poker. Then they delve into poker strategy, then they start pondering poker math. So, with that in mind, I’m going to assume you’ve studied some strategy. Could be wrong, but I like the odds.

So, let’s say you’re holding J-10 of hearts. You have a flush draw. The board is 4hAh4s. You’re opponent was in the small blind. He bet the flop, you seeing your flush draw raised back into him and he re-raised. You counted the money in the pot, the raises were pretty small and there were a lot of limpers preflop that just folded.

What are you putting your opponent on? If he hit only the ace, then you have lost 10’s and jacks as outs. The same goes for any 4. If he has a 4, he has trips, and you still have a flush draw. If he happens to have A4, or 44 you’re actually drawing dead, here with no outs.

Many players are very generous with themselves when counting their outs and calculating their implied odds. I would like to say this can get you into trouble, however, that wouldn’t be entirely accurate. In effort to remain as factual as possible, all I can say is, over calculating your outs will cost you money.