The Math:
How You Too Can Be
a Mathter Poker Player

Math--I hate math. 2 times 2 is 4. (Or is it 2 times 2 are 4? Is that a math question or an English question? Hmmmm.) Now that I have us all sufficiently confused, let’s move on. Why do we need to know math for poker? Ahaa! So we can get the mathematical advantage by making the correct play when the odds are in our favor.

This is how casinos make money; they have a mathematical advantage on every game on the floor. While anyone on any given day can beat the odds if they are lucky, over time the luck factor disappears and the house will make what the math dictates. Some games are better than others, but how do you know unless you know the math?

Playing Baccarat, you will lose only about $1.25 per hundred dollars that you gamble (okay, so it’s really $1.16 on the Banker and $1.32 on the Player, but you get the point--the house has a small advantage in this game, unless you bet the tie. Ouch!). On the other hand, you'll lose as much as $20 per hundred that you gamble on some slot machines (depending on how those particular machines are set. So the house has a monster advantage in this game).

So wouldn’t it be nice to have that advantage when you played poker? Well, you can by doing the math. Here’s how it works:

If the mathematical chances of you hitting a hand are 4 to 1 and the pot is laying you 5 to 1, you’ve got a mathematical advantage over time and you should make the call every time you get the chance to. If, on the other hand, your hand is 4 to 1 and the pot is laying you only 2 to 1, then you have a mathematical disadvantage over time and you should never call.

Pot Odds

First, you need to figure what kind of “price” the pot is laying you. This is easy. Just add up all the money in the pot and divide it by the amount of the bet you have to call. For instance, let’s say the pot has $100 and you have to call a $20 bet. 100/20=5. You are getting 5 to 1 pot odds.

If you think you have the best hand, it’s a no brainer. However, if you think that you must improve to beat your opponent, you have to figure out the mathematical chances of doing that.

Hand Odds

In order to figure the odds of improving your hand, you must first count your “outs.” An “out” is a card that will turn your hand into the best hand.

For instance, let’s say you have pocket tens and your opponent has pocket Aces. In order to beat him, you must catch another ten. How many tens are left in the deck? Well, there were four to begin with and you have two, so that leaves two left in the deck. Therefore you have two outs.

Here’s the way a lot of players figure out the odds: let’s say that after the flop you have two outs. We can see what five cards are: the three on the board and the two in your hand. This leaves forty seven cards that are unknown. So, we have two chances in forty seven. Two cards will win for us and forty five will lose for us. The odds are 45 to 2 or 22 ½ to 1 against hitting one of those cards on the Turn (about 11.25 to 1 with two cards coming--the turn and the river).

Let’s say you have a flush draw. There are thirteen of your suit to start with and you know where four of them are (two on the board and two in your hand). This leaves us nine “outs” (13-4=9). That means we have nine chances in our forty seven unknown cards, or one chance in 5.22, which would make our odds 4.22 to 1 against hitting one of those cards on the Turn (about 2.11 to 1 with two cards to come).

Wasn’t that fun? But is it accurate? Probably close enough. (Did I mention that I hate the math?)

Alrightythen, that’s one way, but the real mathematicians use this handy-dandy formula to compute the percent and odds (and to melt the brains of mathematically challenged poker players). With two cards to come, here’s how to figure out the percentage probability:

Percent = 1-[(47- Outs) / 47) x (46-Outs) / 46) ]

Then they take that percent, magically turn it into a whole number, and divide it into 100, then subtract 1. That gives them the odds. Wasn’t that easy? (Did I mention that I hate the math?)

With just one card to come, the formula is much simpler: Percent = (46-Outs) / 46. Then, once again, we magically turn that percent into a whole number (just move the decimal point two places to the left) and divide it into 100 and subtract 1. What could be easier? (Brain surgery, climbing Mount Everest, and learning a foreign language in a week would be on my list.) Maybe it’s not that bad. Let’s try it, shall we?

Here’s our two “Outer” example from before:
Percent = 1 – [(47 – 2) / 47) x (46 – 2) / 46)]
= 1 – [(45 / 47) x (44 / 46)]
= 1 – [0.957 x 0.956]
= 1 – 0.914
= 0.086
= 8.6% chance with two cards to come (turn and river)

Now turn that % into odds:
100 / 8.6 = 11.63 – 1 = 10.63 to 1

Of course, if you round off to the one hundredth instead of the one thousandth, it makes a big difference. You get this:

Percent = 1 – [(47 – 2) / 47) x (46 – 2) / 46)]
= 1 – [(45 / 47) x (44 / 46)]
= 1 – [0.96 x 0.96] (Here’s where we did our rounding off.)
= 1 – 0.92
= 0.08
= 8 % chance with two cards to come (turn and river)

Now turn that % into odds:
100 / 8 = 12.5 – 1 = 11.5 to 1

To make this easier, you can simply take a calculator, ask your neighbor what the odds are (I’m sure they’ll help), or use your Phone-a-Friend. (Did I mention that I hate the math?)

Praise be to the Poker Gods! There is an easier way to compute the percentage. I call it “The Double, Double.” Double, double is less trouble! (That could make a nice little ditty. Let’s see…Double double, ditty ditty, dom ditty do; I’m hers, she’s mine, wedding bells are going to chime—hmmm, maybe that’s been used before—never mind.)

Here’s how “Double, double” works: take your number of outs and double it. If you have 2 outs, your percent becomes 4. 5 outs would be 10 percent. Man, that was easy. This gives you the approximate percent you will hit on the next card.

In order for you to figure out the percentage with two cards to come, simply double again. Hence the Double, double ditty. (Cool, something I can do without using my Phone-a-Friend.)

To reiterate (I love repeating myself and using the word reiterate. I love repeating myself and using the word reiterate.)

Let’s try this on our two “outer”: 2 times 2 is 4. (are 4? Oh no, not that English/Math question again.) So you’re at 4% to hit on the next card. With two cards to come, you double again. You have an 8% chance to hit your card with two cards coming.

Double, double: easy to do, easy to remember. The bummer is you still have to convert that into odds. To get your odds, take that percent number and divide it into 100 and subtract 1. For our example, you get 12 to 1 with two cards to come or 24 to 1 with one card.

Believe me, this is way close enough for you to make your determination. If you had a flush draw, you would be 36% with two cards to come or 18% with one card to come. So you would be 2.25 to 1 with two cards to come, or 4 ½ to 1 with one.

A lot of you folks out there in the poker world can do this easily in your heads. Good for you. I’m sorry, but I’m not that smart. So what did I do? I cheated; I memorized them.

Here’s a handy-dandy little chart:

Odds with two cards to come; this is the column I memorized. Then I just double it for one card
12 to 1
24 to 1
7 to 1
14 to 1
5 to 1
10 to 1
4 to 1
3.2 to 1
2.6 to 1
2.2 to 1
1.9 to 1
1.6 to 1
1.4 to 1
1.2 to 1
1.1 to 1
.95 to 1
.85 to 1
.75 to 1
.66 to 1

The first column is your number of outs, the second the odds with two cards to come, and the third is the odds with one card to come. (Notice that the third column is just the second doubled.)

Double, double is less trouble! So remember to take your hand odds and see if the pot odds are giving you the right price to call. But is that all you have to consider? No! There’s implied odds too.

Implied Odds

Implied odds are like the second cousin to pot odds, except they don’t come from Tennessee and they have a few more teeth. Implied odds consider the likelihood of you making more money on future rounds of betting based on the probable actions of your opponents and the money in their stacks. So, ask yourself, “What am I likely to extract from these players if I make my hand?” Then add that to the pot odds and see what you get.

For instance: A man, a priest, and a Jewish rabbi walk into a bar…wait, I think I was looking at the wrong notes, er, here it is:

Let’s say you’re playing 5-10 No-Limit Hold’em. Four players limped into the pot before the flop. You have A 5 of spades. The flop has two spades, giving you the nut flush draw. There is $45 in the pot. The big blind bets $30.

You know that the hand odds against improving on the next card are about 4 to 1. There is now $75 in the pot. You are only getting about 1.7 to 1 on your call. So the pot odds dictate a fold.

But what are the people behind you most likely to do based on their calling patterns? If you call, there’s going to be $105 in the pot. The person next to act is now getting 3.5 to 1. Will that entice him into the pot? From what you know about him, you think that he will likely call.

So you call and he calls the $30, making it $135. Now the last person to act is getting 5.5 to 1. If that person is on a flush or a straight draw, he already has the right pot odds to call. He does. If this happens, you're getting 6.5 to 1, a great price to try and make your draw.

Before the Flop Odds

Here’s some other important stuff that you should just memorize:

1. A pair vs. two lower cards: The pair is about a 5-1 favorite.
2. A higher pair vs. lower pair: The higher pair is a 4.5-1 favorite.
3. A pair vs. one high card and one low: The pair is a 2.5-1 favorite.
4. Two higher cards vs. two lower cards: The high cards are a 1.7-1 favorite.
5. A pair vs. two higher cards: The pair is a slight favorite, about 55%. Now that your brain is melting out your ears, my job here is done.

So what have we learned here today?

1. Know the mathematical odds of a pot.
2. Know the pot odds and implied odds so you can judge whether to call or not.
3. Know where the cocktail waitress is so she can clean up your brains, which melted out of your ears already. (By the way she’s in the bar with the man, the priest, and Jewish rabbi.)

In my book, there are a lot of skills more important than the math (and, hey, this is my book!) Some are hand selection, when to value bet, reading your opponents, and knowing how much to overtip the cocktail waitress for cleaning up all those melted brains.

Be sure and read my next article, “Trusting Your Gut.”

So until next time, remember the number one thing in poker is to have fun and enjoy it. It’s not whether you win or lose that counts; it’s whether I win or lose.


Dr. Hope, J.A.P.D
(Just A Pretend Doctor)

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