Pot Odds, Calculating Outs and the Rule of 2 and 4:

This question is asked quite a bit on this forum so I thought it would be useful to have the oft repeated information in one post.

The most sited example is the 4 to a flush, so I will be lazy and use it here as well. You have two cards of one suit, and there are 2 cards of your suit on the board. Another card of that suit would give you the nut flush and the winning hand. What are your odds to draw to the flush? Furthermore, should you draw to the flush (are you getting proper odds)? You have:

K 8

and the board is:

Q A 2

There is $80 in the pot. The bet to you is $20.

The first part, determining the pot odds, is easy. There is $100 in the pot ($80 pot + Villain's $20 bet) and it will cost you $20 to make the call. Dividing the total pot by the price to call ($100 / $20) gives you pot odds of 5 - 1.

Now what about our card odds? Assuming only the flush will give you the winning had (in this example let's say your opponent has at least a pair of aces so we cannot count the three Kings as outs), you have 9 outs to improve, since there are 13 spades in the deck, 13 minus 2 on the board minus 2 in your hand = 9. How do we determine what our odds are of making the nut flush on the flop? Here is the equation:



P is the percentage, UC is the number of unseen cards, which in this case is 47. (52 total cards minus 2 cards in our hand minus 3 cards on the board). Our chances of making the nut flush on the turn are therefore:

1 - (47-9)/47 = 19.15%

Then we turn this into a ratio to compare it to pot odds with:

R = (1/19.15)*100-1 = 4.22 – 1 dog

So, we’re getting 5-1 on our money by calling the bet to us, and we only need 4.22-1 to make the call profitable, so we call.

But what if we’re facing an all in bet on the flop? Villain bets $20, we raise to $60, and Villain shoves in his remaining $140. The pot is now $300 and it costs us $80 to call. So the pot is giving us an immediate 3.75-1 odds ($300/$80). But what are our odds of making the flush? We will see two cards, the turn and the river. So we need to multiply our turn and river odds together using the following equation:



Chug away at that and we get a 34.97% chance to make our flush, or 1.86-1 odds. Our pot odds are 3.75-1 so we call.

Ok, ok. Wait! How do I possibly do all this complex math in my head, you ask? This is where the Rule of 2 and 4 comes in. The Rule of 2 and 4 will give us a good estimate (but not exact) of the percentage chance of making our hand. Keep that last point in mind. These are the odds for making your hand, not winning the pot!

So, to use the Rule of 2 and 4, we multiply our outs by 4 on the flop, or by 2 on the turn and we will get a rough estimate of the percentage of making our hand. On the flop, we had 9 outs. 9 x 4 = 36%, which is roughly the same as the actual percantage of 34.97%. On the turn, we multiply our outs by 2 and get 18%, which is roughly the same as the actual 19.57%. Keep in mind that you can misapply this rule if you are not in an all in situation on the flop but still use the rule of 4 to determine your odds. This is because you will most likely be facing another bet on the turn and, therefore, will not be using the correct pot odds to make your flop decision. Also, the rule of 2 and 4 is not very accurate in situations where you have lots of outs. As you can see, if we had 18 outs on the flop, 18*4 = 72%, where actually you only have 62.44%. But with so many outs, all that really matters is you're a huge favorite in the hand.

Another way to keep these odds straight is to just print out a list like the one below and keep it next to your computer for a handy reference:



Hope this helps. Good luck at the tables!

This is good stuff, thanks.

Very nice post.

A little algorithm for computing estimating the required pot odds from the
probability that you computed (either exactly or using the rule of two or four):

Using your example:

Suppose you have a FD, or 9 outs on the flop. To see if you have to pot odds to call to see the turn....using the rule of 2 we have a probability of 9x2=
18% or a probabiity of .18. Express that probability in the form 1/x. Thus
.18 = 18/100 = 1/5.55555. Then just subtract one from x and you have the odds necessary....in this case 4.55555. The approximation algorithm is thus:

1) Take 2 times your number of outs (in FD example: 2 x 9 = 18)
2) Divide this number into 100 (in FD example 100/18= 5.555)
3) Subtract one from this number (in FD example 5.2222-1=4.555)

This algorithm can be made exact. Instead of dividing 2 times the number
of outs into 100, we instead divide into 94 (two times the number of
unknown cards)....we get the exact algorithm:

1) Take 2 times your number of outs (in FD example: 2 x 9 = 18)
2) Divide this number into 94 (in FD example 94/18= 5.222)
3) Subtract one from this number (in FD example 5.2222-1=4.2222)

which matches the OP's original pot odds. An exact algorithm could also be
given for the all-in on flop scenario to calculate pot odds to river; but I
doubt that it would be very useful.

Thanks! I'm glad you liked it.

The rule of 2 and 4 is not meant to be exact. Rather it is meant to be a quick, and more importantly, easy calculation to do in your head while you're at the table. The point being that you don't really need exact odds to make the correct decision. Anything close will be fine when you're in the middle of a hand.

If you want to use exact odds, the formulae given above the rule of 2 and 4 will do the job. Also, the odds listed in the chart are the exact odds (calculated with said equations) for the given outs (assuming you have counted your outs correctly).

Hi Shick
Thanks for the information. I want to post something more here. Check out Harrington On Cash Games: Volume 1 - Page 38...

Solomon's Rule:
With two cards to come, multiply the number of outs by four, then subtract the number of outs in excess of eight to get your winning percentage.

With 14 outs, for instance, your winning chances are approximately 50 percent.
50=(14)(4)-(14-8)

Some people will not want to or not be able to do this math while playing for others, this can be of assistance. Whether or not someone will use Solomon's rule will depend on their amount of time to act, patient, skill, and motivation. Good luck, Yojimgari

To figure out your odds of improving, just memorize that chart. You don't have to memorize the whole chart. In Texas Holdem, I use pot odds only when I have between 6 and 12 outs. After playing 20k+ hands, it should become second nature.

For years I have simply used percentage odds and percentage of pot, instead of ratio odds.
Here's the summary:

Traditionally, you'd do something like this - the pot has $40 on the turn, villain bets $10. I have to call $10 to see river. Thus my pot odds are $50:10, or 5:1 ($50 because his bet is part of the pot, of course). So I need a draw that has better than 1:5 odds (16.67%) in hitting to be making an acceptable drawing call.

Percentage-wise, I would think of it like this - out of what will be a total pot of $60 (with villain's bet), $10 is roughly 16%. So a draw needs to be 16% or more likely to hit to make a call acceptable. On the turn, with 36 unknown cards, this equates to 5.76, i.e., 6, cards, and, in ratio form, 6 cards compared to the 30 other cards is 1:5.

Does anyone see anything wrong with this method? From my point of view, the only disadvantage is the rule of 2/4's inaccuracy with very small or large number of outs (for a low number you have to add up to a few percentage points, for a high number, you have to subtract some - the latter being very problematic in Pot Limit Omaha, for example).

But the much easier accessibility of this method has made it stick with me. I suppose if I only played limit games, I might find it easier to track big bets in the pot, but playing NL 50% of the time or more, I find that it's just not possible to always have any running count of units in the pot except for the rough total pot tally.

Anyway, I am mainly curious to see if anyone else has any thoughts - either problems with the percentage technique, or reasons why they have come to prefer the traditional odds method using the ratios.

No, there's nothing wrong with using percentages instead of ratios--as long as you're figuring them correctly. They are both expressing the same thing, just in a different way. If you read any poker books or watch any math videos on poker sites, they will usually have you converting your results into ratios and comparing them that way. As you can see, the chart includes both, so you can just look up your outs and find the correct unit.

It's pretty accurate with small numbers of outs. Only about 1/2 of a percent low with 2 outs to come.

For high numbers of outs, you can use Dan Harrington's "Solomon's Rule" as described by Yojimgari, or just use the chart. You'll be looking up the same number of outs so often, it won't be too terribly long before you have them memorized for all your typical draws.

Nice thread... I love reading about this stuff. I need to develop a habit of calculating pot odds with ease every hand. For now I'm going to use the computer calculator I'm going to calculate almost every hand and maybe one day all the calculating will be stuck in my head. I feel my poker brain growing everyday(ow, it hurts).

Just a minor geeky correction...

What were actually doing with these probability equations if determining the probability of not hitting our outs and then taking the additive inverse. My apologizes for any confusion this oversight may have caused.

A few corrections to your method.

You don't need to convert to a ratio if you're going to just use percentages. You're just adding an unnecessary step. Convert the pot odds directly to a percentage with: ($ to call) = x% * ($ in pot) and solve for x.

In your example, this would be 20%. (10 = x% * 50, solving for x = 20%).

see above.

Using the rule of 4 (or a quick look at the chart) and we see we need 5 outs or more to make a profitable all in call on the flop (2 cards to come): 5 x 4 = 20

Thanks for the replies Shick!

I normally don't do the ratio first, I was just calculating the ratio answer to the example first for comparison.

I do see that I miscalculated the percentage though because the total pot should be $50, so $10/$50 = 20%, as you say. I had included my $10 call as part of the total pot.

Actually, and perhaps now I am confusing myself - but if you're calculating odds in percentage form, don't you need to include your call as part of the total pot? Because the pot odds in ratio are 5:1, so to convert that to percentage, you would do 1/(5+1) = 1/6 = 16%!

Actually, come to think of it, I usually DO include my call of the bet as part of the total pot (as I did in my original example). This does make a big difference in no limit - for example, let's say you are on the turn with an open ended straight flush draw, the pot is $100, and the villain bets 1/2 pot for $50. The way I would calculate it is I have an easy call, because the total pot if I call will be $200, and I am contributing $50 of that, or 25%. With my 15 outs, I have 30% chance to hit my draw on the river.

Am I wrong, crazy, or both? Have I really been miscalulating my pot odds percentage-wise, or is that right to include your own call as part of the total pot that you will divide your bet into to get the percentage?

Yeah, I was just thinking about this myself. You're completely correct. When you're figuring the percentage, you need divide your call by the total pot including your call. In other words, in your example the pot was $50, $10 to you. So the percentage is 10/60 = 0.16666 or 16.7%.

As opposed to figuring your pot odds which are ($ in pot)/($ to call) or $50/$10 = 5:1. Here we don't include our call in the pot figure. If we convert 5:1 to a percentage, however we get the same 16.7% (we win 1 in 6, or 1/6).

So, yes, you can use a percentage to figure your pot odds, just make sure you're figuring it right

Simple

(call $) : (total pot)
(outs) : (cards left - outs)

You have flush draw. Pot is $40
Villian bets $10 (pot now $50)
call : pot
$10 : $50 (1:5)
outs : cardsleft-outs
9 : 38 (47-9)
so
1 : 5 to 9 : 38 (divide 38 by 9 to get 1 : 4.22) (even as a rough calc in your head, you should say 4.2 to 4.25)

I guess it is all different ways of doing it. But the least number of steps possible to get the most accurrate. You really don't need to keep converting numbers back and forth from % to ratios.

/my2cents

Nice post!

Yes, to put your equation in terms of the OP, we get:

P = (UC - O)/O

Where solving for P gives us the odds to 1 against, UC is unseen cards, and O is # of outs.

So, assuming we're not in an all in situation (i.e. we're only calculating our outs for one street) and we can do some quick long division in our head, this works great. However, you can't multiply ratios together to get your odds for an all in call with 2 cards to come, which is why we do the calculation as in the OP.

In the OP we solved for the additive inverse which yields a percentage. Multiplying the percentage chances of hitting our card on the turn or the river, then taking the additive inverse yields the percentage chance of hitting our outs on both streets.

We are essentially doing a rough approximation of this with the rule of 2 and 4.

Yes, to put your equation in terms of the OP, we get:

P = (UC - O)/O

Where solving for P gives us the odds to 1 against, UC is unseen cards, and O is # of outs.

Hi im really struggling with the maths can u please just write what the abbriviated letters stand for

Thanx

forget that ive found it was just being lazy

okay but what I dont get is how do I know how much percentage that I need to call the size of the bet my opponent is making?

guys cmon' I really cannot catch on to this for some reason and I really want to get this down... I think it would really help my game a ton to learn the math of poker!! Please help!!

I need help understanding this please!!

I think what you're asking is at what odds does making a call become profitable. Basically, if the odds of making your hand are less than the odds the pot is laying you, then it is a profitable call. For instance, if you have odds to make a flush of 5:1 and the pot odds to call are anything greater than 5:1, then calling puts you in a profitable situation.

what does FD Mean? That's another thing ... Twoplustwo.com has an abbreviations link but it only covers about 40% of the lingo ... anyone know where I can find further material that will help me understand what players are talking about when they used acrynomns or abbreviations like OP LM GTG FD and stuff like that?

ok thank you shick but do you think you could give me a quick hand example? cause I am not sure how to figure out if the pot odds to call are greater that 5:1

As stated in the OP (Origional Post):

Pot odds = $ in pot / $ to call