pot odds question

Hello, I'm fairly new to poker. I understand the basic idea of pot odds but I don't understand the justification.

For example, suppose you're heads up with an opponent and have an Ax suited and you have a flush draw on the turn with $100 in the pot and no tainted outs. Also, assume our opponent is really clever and knows if you hit your flush or not, so he will not call any raise if you complete your flush. Finally, assume this flush will be the nuts. So you have a 19.57% chance to win the pot. According to this idea of pot odds, you should not call a bet larger than 19.57% of his bet plus the pot. So you should only call if his raise is less than $24.33. Calling anything over $24.33 would give you "bad pot odds." However, if you look at your expectation...

EV = 0.1957*(124.33) + (1-0.1957)*(-24.33) = 4.763 > 0

So you could still call a raise greater than $24.33 and have a positive expectation. In fact, if you call any raise less than $32.14, you will still have a positive expectation.

In general, suppose we have a probability p of winning a hand and a bet b which is a fraction of the pot. Since the bet is a fraction of the pot, then the pot will be 1. So our expectation is

EV = p*(1+b) + (1-p)*(-b) > 0.

Assuming that p < 1/2, then we have

b < p/(1-2p).

The strategy of pot odds outlined on this site, and most other sites, states that you must only call bets that are less than p*(1+b) of the pot. So b < p*(1+b), giving us

b < p/(1-p).

Assuming we use this strategy, then b < p/(1-p) < p/(1-2p) whenever p < 1/2. So we're going to fold to bets that still give us a positive expectation.

So my question is why should I fold these hands to these bets when I still have a positive expected value?

To be honest, I have no idea about your math. I was never any good with math. I think we're gonna have to wait for koolmoe to help us with this one.

But to address another issue. Playing Axs and chasing a flush, you will always have tainted outs, unless you know what the other guy is holding. If the board pairs one of the non suited cards, it may complete your flush, and someone elses full house.

I am basically a lame duck when it comes to stats but here are my comments....

The quick answer is that you really dont ever know the EV of the hand b/c of all the other unknowns so the Pot Odds rule is fairly solid and quick to figure out....

Although, I believe you are correct in that you could probably call a bet that is a bit higher than 20%, is your expected value is positive. The pot-odds rule of thumb is very good in deciding to call b/c it gaurantees a profit and its easy to calculate quickly when you have to make a decision. You also have other hidden factors which can negatively impact your expected outcome (e.g. tainted outs, raises, non-nut hands, implied odds) all of which probably makes it very complicted to figure-out the actual expected outcome. Maybe the pot-odds rule of thumb puts enough cushion over the EV that it accounts for all the other unknowns and solid for making betting decisions on.

pondering this..

Hi benthehen, I think I see an error in your argument. You state that EV in your hypothetical scenario is:

EV = 0.1957*(124.33) + (1-0.1957)*(-24.33) = 4.763 > 0

but the real EV equation is:

EV = 0.1957*(100.00) + (1-0.1957) * (-24.33) = 0

When you win the $124.33 pot, you must subtract from your winnings the $24.33 which you had called, because you didn't gain it; you had it to begin with.

Originally Posted by mikes007

When you win the $124.33 pot, you must subtract from your winnings the $24.33 which you had called, because you didn't gain it; you had it to begin with.

I think in this example the $124.33 is the $100 + the $24.33 bet the opp bet. If you call the pot would $168.66 and on the river (if flush hits) you would profit $124.33. If this example is correct, there is definately positive EV.

I guess the overall comment is that if your call is equal to pot odds, you should expect a positive but minor EV in the long run. Also, I think in a normal situation, some of the money in the pot should be yours to being with, which could drastically decrease the EV of the results. I this example, the $100 was not from the caller

But then this is a misapplication of pot odds. You have to add the amount of the bet and the call into the pot when calculating the odds. I thought the same thing as you, Krapp, until I really started thinking about it. Eg pot is $1. Opponent bets $1. My pot odds are 1/3 if I call $1. Assuming I have a 33% chance to win:

EV = (1/3)*2 + (2/3)*(-1)

EV = 2/3 + (-2/3) = 0

As far as some of the money in the pot being yours to begin with, that is EV irrelevant. We are only concerned with money from the future. not the past. The reason we need to subtract the call from the winnings in the EV calculation is because it is dependent on the decision we make; it is NOT already in the pot.[/quote]

mikes, according to the FTR pot odds essay, you don't include your own call as part of the pot when calculating pot odds.

to take a different tack, how relevant are pot odds heads-up?

In a tourny or cash game? In a tourny I'd say useless becuase it's impossible to factor in the value of busting him with a certain hand, so it would be hard to calculate. In a cash game, however, if you feel you need some help justifying a call or fold I'd always reference pot odds but other than that, i'd ignore them.

-'rilla

Originally Posted by benthehen

So you have a 19.57% chance to win the pot. According to this idea of pot odds, you should not call a bet larger than 19.57% of his bet plus the pot.

You are misundertanding odds versus probability. Convert the 19.57% probability to odds and you get approximately 4:1 (one in five is four to one). At the time you call, the pot should be at least 4 times as big as your bet. Simply put, four times you will lose your bet, and one time you will *profit* 4 times your bet.

124/4 = 31 (obviously fine to call $24 here)
132/4 = 33 (getting close to even money to call $33 here)

If you want to multiply the 19.57% by the pot size, you need to include your bet as well (19.57%*148 or 19.57%*164, as the case may be).\

Convert the percentages to odds, memorize them, and just start counting bets in the pot (easier than trying to multiply in your head).

Clear as mud?

Originally Posted by TheNatural

mikes, according to the FTR pot odds essay, you don't include your own call as part of the pot when calculating pot odds.

Unfortunately, that explanation is incorrect. You can call a pot-sized bet with a nut flush draw if you could see both the turn and river with no further betting.*

Example:

$10 pot, opponent goes all in for $10, you call with Axs and two of your suit on the board. You call.

You are a 1.86:1 underdog. 186 times you lose $10 = -$1860. 100 times you profit $20 = $2000. Net win is $2000-$1860 = $140. Profit per hand = $140/286 = about $0.50 per hand.

It's a wonder Tyson ever posts a profit. :P

*A common mistake for beginners is to ignore the betting that will occur on the turn. There is only a 4:1 chance for them to complete the flush on the turn, but they will often call a 1/2 pot bet getting only 3:1 on their money. If a blank falls on the turn, you can then bet them out of the pot yet still make profit from their bad call on the flop.

Ok, after reading Koolmoe's and mike007's posts about 10 times, I think I understand. I hope I'm not going to make a fool of myself so here goes..

Basically, the problem with the original post is the figure of 24.33 is calculated incorrectly.

When caluclating pot odds, the "pot size" is considered to be all the money that will be in the pot after you call the bet. It turns out that if the pot is currently $100, the max bet that your opponent could make that you could call would be about $32, not $24.33. I just figured that out through trial and error. If the pot was $100, and one person bet $32 and you call, then the total "pot size" is $164.

(.1957)*$164 = $32.

So in your EV equation, if we put in the values of $32 (the amount you would lose about 80% of the time) and $132 (the amount you would win about 20% of the time), it works out.

EV = 0.1957*(132) + (1-0.1957)*(-32) = about 0


So.. if I'm understanding this correctly... Tyson's pot odds essay is wrong? He doesn't include his own call as a part of the total pot size. Is this what you're saying koomoe?

I'm cool with my 35%, thank you

When caluclating pot odds, the "pot size" is considered to be all the money that will be in the pot after you call the bet. It turns out that if the pot is currently $100, the max bet that your opponent could make that you could call would be about $32, not $24.33. I just figured that out through trial and error. If the pot was $100, and one person bet $32 and you call, then the total "pot size" is $164.

Your right, the maximum call is $32.14. You can use B < P * p/(1-p) to find the maximum call for positive expectation. Where B is the bet you must call, P is the pot, and p is your probability of wining. However, I keep reading online that $24.33 should be the maximum call and I don't understand their justification.

Here is an explanation of pot odds that I found on the web: http://www.pokertips.org/glossarydefs/10.php

According to this site, the following must hold:

(pot + bet) * (chance of hitting) >= bet

Or in terms that I was using earlier:

(1+b)*p >= b

Giving us b <= p/(1-p). In my previous post, I argued that there exists bets larger than p/(1-p) that you can call and still have a positive expectation. In other words, there are bets that your "pot odds" would not justify but your expectation would justify ("pot odds" according to the link above.)

To illustrate with an example, suppose there is a pot of $1, and your friend makes a bet B that he can win a coin toss. If you win, then you get his bet B plus the $1. If you loose, then you loose your bet B. Since you have a 50% chance of winning, then it is clearly a good call no matter how large the bet (assuming you can afford to lose quite a few bets.)

However, when p = 1/2, then b <=p/(1-p) = 1. Remember that b is a fraction of the pot. So according to link above, we should not call our friend's bet if it is greater than $1, which is an apparent contradiction.

So is there any reason for this or did the author just screw up? Or maybe I screwed up?

Originally Posted by TheNatural

So.. if I'm understanding this correctly... Tyson's pot odds essay is wrong? He doesn't include his own call as a part of the total pot size. Is this what you're saying koomoe?

Tyson's essay is incorrect.

The standard way to use pot odds is to calculate the probability that you will win (make your draw in this case) and convert that probability to odds by dividing the probability you won't win by the probability you will win.

In the case of a flush draw with one card to come, the probability of making the draw is 9/46 = 19.57%, so the probability of not making the draw is 100% - 19.57% = 80.43%.

That means the odds are 80.43:19.57, which is about 4.1:1. The 4.1 is the number you use to determine the size of the pot (meaning you can call up to about 24.33% of the pot).

Odds compares the amount you bet versus the amount you profit. You call $32 to win $132, for example (Notice that 32*4.1 = 131.2).

Probability multiplies the total amount of the pot after all bets are put in. 19.57% of the time you win $164, for example, making the value of the pot $164*.1957 or $32.

Notice that properly doing the calculation using probability would be difficult at the table. That's why most people convert the probabilities to odds for use at the table.

Originally Posted by benthehen

Here is an explanation of pot odds that I found on the web: http://www.pokertips.org/glossarydefs/10.php

According to this site, the following must hold:

(pot + bet) * (chance of hitting) >= bet

The site is incorrect. Incidentally, they've also missed the probability of hitting the flush on the river (it's 9/46 or 19.57% - not 18%).

I guess I always have been in the habit of not including my bet in calculating pot odds since I dont want to make it a habit of calling down bets that have zero EV. Afterall, what point would there be if I just broke even in the long-run

Krapp, I think if the bet is truly 0 EV per the pot odds you probably still want to call because of the implied odds. The assumption that your opponent is a poker genius and will automatically fold if your draw hits is not realistic.

Also, this knowledge is helpful when you're trying to give your opponent incorrect pot odds to call.

I want to thank benthehen, mikes007, and KoolMoe for all the analysis. I have definitely learned something here that will help me at the tables.

Originally Posted by koolmoe

Originally Posted by benthehen

Here is an explanation of pot odds that I found on the web: http://www.pokertips.org/glossarydefs/10.php

According to this site, the following must hold:

(pot + bet) * (chance of hitting) >= bet

The site is incorrect. Incidentally, they've also missed the probability of hitting the flush on the river (it's 9/46 or 19.57% - not 18%).

Why is that site incorrect? (other than the 18% thing). I think by "pot", they mean all the money in the pot, except for your call. And by "bet", they mean the amount you have to call.

man you people should give this site a break! it's a good site, and that 18% clearly states ROUGHLY 18%. they're just using the 2% per out.

Originally Posted by TheNatural

Why is that site incorrect? (other than the 18% thing). I think by "pot", they mean all the money in the pot, except for your call. And by "bet", they mean the amount you have to call.

I thought that also, but look at the last example, where there is $10 in the pot before the bettor bets $40 into you. They compute that your pot equity from calling the bet is ($10 + $40)*18% or $9 when it is actually ($50 + $40)*19.57% or about $18. You still can't call the $40 (they got that right), but the process they use is incorrect.

I'm glad that I could help in this discussion so soon after starting to post. It's good that everyone has been able to analyze the question and confirm what I thought, because most sites have a confusing or erroneous explaination of pot odds. I'm happy I found FTR, the forums are excellent.

Originally Posted by koolmoe

Originally Posted by TheNatural

Why is that site incorrect? (other than the 18% thing). I think by "pot", they mean all the money in the pot, except for your call. And by "bet", they mean the amount you have to call.

I thought that also, but look at the last example, where there is $10 in the pot before the bettor bets $40 into you. They compute that your pot equity from calling the bet is ($10 + $40)*18% or $9 when it is actually ($50 + $40)*19.57% or about $18. You still can't call the $40 (they got that right), but the process they use is incorrect.

ok.

Originally Posted by TheNatural

Krapp, I think if the bet is truly 0 EV per the pot odds you probably still want to call because of the implied odds. The assumption that your opponent is a poker genius and will automatically fold if your draw hits is not realistic.

Also, this knowledge is helpful when you're trying to give your opponent incorrect pot odds to call.

I want to thank benthehen, mikes007, and KoolMoe for all the analysis. I have definitely learned something here that will help me at the tables.

Implied odds is a very tricky proposition since you are trying to predict what your opp might do in the future and probably only relevant in No Limit Holdem. However, if you are playing No Limit, pot odds (or EV = 0) is significantly less of a factor when dealing with potentially large pots. Although, I believe if you are making betting decisions based on implied odds, your gonna have very streaky results as opposed of making most of your betting decisions on pot odds and > 0 EV situations.

How can you claim implied odds are only important in No-Limit?

One way to look at implied odds is even though a call right now might not be justified by the immediate pot odds alone, you may be able to "catch-up" in an EV sense if you have good implied odds.

Surely this thinking can be applied in Limit as well as No-Limit, No?

There is certainly a *larger* implied odds factor in no-limit ... (i.e. in no-limit, if you hit, you have the possibility of taking your opponent's whoile stack). I don't think anyone can dispute that fact.

I just re-read this thread, and it still seems that people want to include the amount of the bet they are about to call in the pot total for pot odds calculations. That is NOT correct.

*Immediate* Pot odds:
--------------------------
represent the immediate return on your investment (ROI) (whether you end up winning the pot or not).

If you have to invest $5 to win $15 that's already in the pot, then the pot is laying you 3:1 pot odds. Note that your opponent's original bet that you are considering calling of $5 that brought the pot up from $10 to $15 is included in the calculation.

This means you can lose the hand 3 out of 4 times and still have a nil return/loss on your *immediate* investment (i.e. EV=$0). Why? Because the pot is paying you 3:1 for your investment. That one time out of 4 that you do win the pot, you are getting a 3:1 return on your immediate investment thus compensating for the money you lost the other 3 times outta 4.

EV Calc
---------
Scenarios where you don't win the pot, you've lost $5. That will happen 1 out of 4 times since we're assuming you are a 3:1 dog.

The scenario where you win the pot, you make $15 ($20 less your $5 investment). Again, that will happen 3/4 times since we're assuming you are a 3:1 dog.

The correct EV equation for this example would be:

1/4*$15+3/4*(-$5) = $3.75 - $3.75 = $0 = EV

i.e. you break even in the long run

Implied Odds can make an EV=0 situation EV +'ve
---------------
As Jonny mentioned, if you hit your draw and your hand ends up being the best hand, then you stand to make money in the long run if you can get extra bets out of your opponents (i.e. you have implied odds which turns your call from EV neutral to EV positive). That why you would call even though it appears your ROI is zero.

Effective Pot Odds (when you have more than one card to come)
---------------------
Now, where pot odds get more complicated is when you are on a draw and yes, sure the *immediate* pot odds justify a call, but you may be facing having to call bets on future streets if you don't hit on the next card (if you did hit, and you thought you had the best hand you should raise for value since you've got a 'pot equity edge'). This example is where *effective* pot odds kick in. You have to assume that with more than one card to come that you'll have to call bets on future rounds (unless of course, your opponent gives you free card(s) - which is why we all love to play against passive players, or ones that will give you the free turn and/or river cards when you raise on the flop with a draw). So, to justify a call in these situations (I'm assuming your immediate pots odds are equal to your odds of winning for this example), you have to be sure that if in fact you do hit, you're opponent will pay you off with enough $$ more often than not to cover your expenses of calling future bets. Otherwise, in the long run, you won't get compensated for all the money you lost when your draw didn't come in, yet you called one or more bets hoping it did. In other words, in the right situations you can look at it as if implied odds can partially underwrite the risks associated with not hitting your winning hand just like in the above immediate pot odds example.

Also, something else that people don't often consider ... if there's **potential future dead money** in the hand still.. i.e. callers coming along for the ride who have no hope of winning the hand, they are also under-writing your calls. So, you can loosen up a bit with loose callers in the hand and you will stay EV +'ve with marginal calls based purely on immediate or effective pot odds alone. This is where the concepts of 'Pot Equity' and 'Pot Equity Edge' prove useful. In other words, dead money coming along for the ride increases your Pot Quity Edge relative to what it would be by cards odds alone.

I recommend you pick up Theory of Poker (Sklansky), or SSH (Miller/Sklansky) if you want really good explanations of all this stuff.

I'd love to get some feedback on this post to see if people find my explanations clear rather than just muddying the waters more.

Does anyone have any "tricks" for doing the math at the tables where you have to consider effective pot odds with more than one card to come?I'm thinking more about LHE here since in NL you'd have to make assumptions about the size of future bets you'll have to call

Also, can someone give a clear explanation ... ideally with examples, of the concept of 'Fold Equity' ....

Originally Posted by RiverMonkey

I just re-read this thread, and it still seems that people want to include the amount of the bet they are about to call in the pot total for pot odds calculations. That is NOT correct.

It's not an EV calc, it's a pot equity calc. For a pot equity calc, you do have to include your bet for the calculation. Going by expressed odds alone, your pot equity needs to be at least as big as the bet you have to call to continue.

Lots of people do pot equity calcs instead of EV calcs, which leads to the confusion you see in this thread.

I agree with your EV calc in the post above. Notice that if you constrain the EV expression to be greater than zero and solve for maximum bet you can call you have the pot equity calc. Also notice that you wind up multiplying the entire pot (including your bet) by the probability. It's a much easier calc to do at the table than the EV calc. Suppose that p is the probability your draw will hit and win, P is the size of the pot (prior to your call), and B is the size of the bet you must call. The biggest bet you can call profitably based on expressed odds alone is given below.

p*P + (1-p)*(-B) > 0

which implies

B < p*(P+B)

The equivalent pot odds expression is

B < [ p / (1-p) ] * P

Notice that p / (1-p) is the odds of your draw hitting rather than the probability of it hitting.

Thanks KoolMoe ... my intended question (I didn't articulate it well) was:

What's the difference between fold equity and pot equity edge?

I think you answer was: 'they are the same thing'; *fold equity* is just another term for *pot* equity edge.

e.g. say you are a 3:1 dog to win a pot, but there are still 5 people in the hand. You will win more than your fair share; you win all the $$ in pot 1 outta 4 times, and yet you and 4 other people are still contributing new $$ to the pot. 25% of the time you win the pot, but you are only contributing 20% of the new $$ going in ... You have a pot equity edge of 5%, so pump 'er up!!!!!