Basic Poker Math

Poker is a game of skill buried deep within a game of luck. In this article we're going to cover expected value, equity, risk, odds, outs, and all of the other "luck" factors about a poker hand.

Expected Value

Expected value is the amount of money you stand to win or lose when you make a bet. It applies in any gambling situation. It's basic probability: if you're flipping a coin and you wager $5 to win $10, your EV is $5, which is a $0 gain. Your possible outcomes are {0, 10}. These are two options and the average of them is $5. If you wager $5 to win $20, your EV is $10, a gain of $5. Your possible outcomes are {0, 20}. Divided by two, equals ten. If you wager $5 to win $2, your EV is $1, a loss of $4. Your possible outcomes are {0, 2}, divided by 2, is $1.

Now let's talk in terms of dice. If you wager $5 on a six to win $30, your EV is $5 (a gain of $0). Your possible outcomes are {0,0,0,0,0,30}. The average of which is $5. If you wager $5 on a six to win $40, your EV is $6.66 (a gain of $1.66). Your possible outcomes are {0,0,0,0,0,40}. The average of which is $6.66. If you wager $5 on a six to win $6, your EV is $1 (a loss of $4). Your possible outcomes are {0,0,0,0,0,6}. The average of which is $1.

You want to make bets where your EV gain is GREATER THAN $0. If you make a bet with an EV of $0, you're gambling for no gain. If you make a bet where your EV is less than $0, you're gambling for a loss.

Counting Your Outs in Poker

Outs are defined as any card that might come that would give you the best hand, assuming you don't already have the best hand. You usually count outs when you have something like a draw or when you have a good-but-not-great hand and think you may need to improve to beat your opponent. If you think you have the best hand, you don't have to count your outs, you're just going to try to put as much money in as possible.

Hand example #1

Let's talk about the most basic "outs" situation, when you have a flush draw, like on a board of . In this hand, you have to assume you don't have the best hand. If your opponent goes all in, he probably has a pair. He could just have a draw, like the or , in which case you're ahead, but we're not going to consider those situations because we want to talk about counting outs.

First, count how many specific cards, out of 52, would make you the best hand. There are 13 spades in a deck (and of every suit), but on this board, we already see four of them: . That means that there are nine spades unaccounted for. You can assume that, any time a spade comes, you're going to have the best hand. Of course, if something like was to come and your opponent makes a full house, it might cost you the hand, but that's very rare, so it's considered an out. We're going to assume that nine cards would come to give you the best hand.

That is, of course, if you don't think your is an out. If the would give you the best hand, like if he just had top pair, then you'd also have the best hand, so you have some extra outs here. This would add three overcard outs to your hand if top pair would win you the hand. On this board, it's probably enough to win the pot.

With your spades plus the ace, you've got nine outs for the flush and three outs for top pair, but the overcard won't always be an out, so it's prudent to count maybe one and a half outs in this case. That would mean that 50% of the time, your ace is good. A fair assumption, especially since you have a weak kicker.

In this hand, you've got somewhere between 9 and 11 outs. In the next section of this article, we're going to discuss how to convert your outs to equity using the Rule of Four. Until then, I'll just tell you that you're 40% to win this hand. That's really good odds if there's any kind of overlay in the pot. You can tend to get all in if you don't have a really big stack here. You can also opt to raise all in and use your fold equity to add value to your hand.

Knowing how to count outs will keep you from overvaluing your hand but also ensure that you do get the proper value for the hand you do have.

Read 3 examples on counting outs.

Converting Outs to Equity in Poker

Your equity is defined as the percent of the time you're going to win the hand. If you've got a flush draw, you know your equity is about 35%, but how do you get that number? You can use software called PokerStove (watch the video).

Hand example #1

Let's say we have a flush draw. We have the on a board.

We know from the above section on counting outs that we have nine outs for the flush draw and probably three outs for the ace, but since those are partial outs, we'll count two outs for the ace. That gives us a total of 11 outs.

Here's where we learn the Rule of Four. Multiply your number of outs times four. This is the equity of your hand and the percent of the time that you will win if the hand was to get all-in right now.

The Rule of Four says that if we were to get all in right here, we would win the hand about 44% of the time, because we have 11 outs. In fact, when we do use PokerStove to calculate our chance to win, the actual chance is 45%.

Hand example #2

Let's say we have the on a board. We know if we catch a jack we're going to win, but let's say they have a hand like T9 for two pair. We're also going to have to survive a ten or a nine coming on the river. Four outs for the jack time the rule of four is about 16%, but we're actually going to be a little less than 16%.

Simply put, drawing for a gutshot against two pair gives us fairly poor equity. On the other hand, instead of having T9 had something like T4, we'd have a lot more equity, about 40%, because now our kings and queens are both outs.

The rule of four is all you need to calculate your equity at the table, and knowing your equity is good because it tells you what pot odds you need to call a bet.

Pot Odds and Implied Odds

Pot odds are the odds that the pot is laying you to call a bet.

Example: There are 300 chips in the pot, and your opponent bets 100 chips.
If you call, you’ll be putting in 20% of the pot (100 chips in a 500 chip pot).
You can call if your pot equity is greater than 20%. Remember, your pot equity is the percentage that we calculated in the last section.

Implied odds represent the money that will go in the pot after you catch your draw. Calculating your implied odds is a little more involved than calculating your pot odds, but it is one of the things that is crucial to understanding where you stand in the hand.

To calculate your implied odds:
Step 1: Multiply the size of the pot after calling times .6.
Step 2: Multiply this number by a number between .1 and .9 which is an educated opponent-dependent guess and represents his likelihood to bet the next street or call a bet on the next street. A higher number represents a greater likelihood of putting in a bet on the next street.

Example: Consider a 600 chip pot versus a very aggressive opponent who bets 300 chips on the turn.
My pot odds dictate that my draw needs 25% equity to call.
My implied odds are worth 1200 * .6 * .7, or about 500 chips. Instead of risking 300 chips to win 1200, I am risking 300 chips to win 1700. Now, I only need 17.6% equity to call.

Several more of our hands are worth a call once we consider the implied odds! Let’s talk for a minute about the “opponent-dependent educated guess” number. I typically use the following calculations:

Tight Passive Player – 0.2
Loose Passive Player – 0.4
Aggressive Player – 0.6
Maniac – 0.8


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