It’s not the most definitive guide out there, but I hope it’s still useful for you guys. And if it is, I’d love it if you could stumble or digg it, or even just share it with your friends. You can stumble and digg it right from here:

Beginning Poker Math

Feel free to download it, use it for reference or share it with anyone you think would like it. Actually, feel free to make use of it however you want. It’s completely free for download.

A lot depends on the raise sizes. I currently play the microstakes SNGs online, where my standard raise size is between 2.5 to 3 big blinds (depending on how tight the table is), so let’s go with 3 big blinds.

For ease of calculation, let’s assume I’m in late position, it folds around to me, I raise to 3bb, and it folds around to the big blind, who calls. So that means there’s now 6.5 big blinds in the pot, when the flop comes.

The big blind is often going to check to me (the standard “check to the raiser”), and I’ll bet about half the pot here, so another 3.5 big blinds. That means I’ve placed a total of 6.5 big blinds in, and the pot is now 10 big blinds.

For a start, let’s assume that I can only win by my opponent folding. That means, if the probability of my opponent folding is x, then, for it to be a good play,

x * 10 – 6.5 > 0
x * 10 > 6.5
x > 6.5 / 10
x > 0.65 = 65%

Therefore, I need my opponent to fold at least 65% of the time on the flop, for this to be profitable. This is about equivalent to him folding every single time he does not pair the flop.

However, this also assumes that we never hit the flop ourselves. Let’s say I have a hand like AJ suited. I then have more than 13.75% chance of hitting TPTK or better, as discussed here. If we use that 13.75% (which, as one of the commentors pointed out, is in actual fact, a low estimate; it discounts the flops when you pair your Ace, and also discounts the times when you flop mid pair, or two overs, or draws) as the times we will win the hand, we then have the following.

x * 3.5 + (1-x) * (0.1375 * 13.5 – 6.5) > 0
x * 3.5 + 0.1375 * 13.5 – 6.5 – 0.1375x * 13.5 + 6.5x > 0
3.5x + 1.856 – 6.5 – 1.856x + 6.5x > 0
(3.5 – 1.856 + 6.5) x > 6.5 – 1.856
8.144x > 4.644
x > 4.644 / 8.144 = 0.57 = 57%

Meaning that your opponent would have to fold at least 57% of the time.

Let’s look at the co-relation of how often your opponent will have to fold (which we’ll represent with x), with how often you have to win if he calls (y). To break even, therefore:

x * 3.5 + (1-x) * (y * 13.5 – 6.5) = 0
3.5 x + 13.5y – 6.5 – xy + 6.5x = 0
10x + 13.5y – xy = 6.5
y(13.5 – x) = 6.5 – 10x
y = (6.5 – 10x) / (13.5 – x)

For a few values of x, we have the following table:

So, what does this show. Even if your opponent does not fold at all, you still need to win less than half the time to be profitable.

More realistically, though, you’ll win the pot about 10-25% of the time when called on the flop. I won’t go into the details of that here, there are too many possibilities to go through, but if you consider 13.45% the chance you flop TPTK with a J-high flop, or 2-pair or better, and add in the 9% of the time or so that you flop an A and no J, Q or K; as well as the semi-bluffs you may hit with if you flop and bet on a draw, I think 10-25% is not too much of a stretch.

Given that, that means you need your opponent to fold between about 30% to 50% of the flops. More specifically, 32% to 52% of the flops. If you average that to 42%, that means he needs to be “bluffing”/semi-bluffing the flop about 1/3 of the times he misses.

Whether this is profitable or not really depends on the player. But from the (admittedly limited players) I’ve encountered online, I think it’s not unrealistic to expect them to fold to about 35-40% of your c-bets, which would mean you need to win the hand about 23-27% of the time when they call. If you’re not playing too wide a range, that’s highly possible, I think (that’s about the chance of you hitting one of your cards on the flop).

Of course, any strategy that’s too obvious can be exploited, and any time you’re unable to adjust to your opponents, you’re losing out. But in general, I think it’s not a bad default strategy to fall back on, especially against unknowns. I think the required fold equities are highly possible, and with Poker Tracker, you can keep track of how often a player folds on the flop, and use that to make a decision as to whether this is a good strategy. I think, based on the above calculations, if he’s folding to at least 35-40% of flop c-bets, it’s a profitable strategy to use.

What do you think? How often do you c-bet your flops, and how much?

Photo by openbox

I’ve compiled some of what I feel are the most relevant of these odds into one short/simple ebook. It includes some information on starting hands, comparisons of hand types, among other things.

And I’m giving it away for free for anyone who signs up to get the free email updates for this blog, so if that sounds good to you, sign up now (it’s free)!

So, in 2 week’s time, I’m giving away a free poker book, of your choice (anything below $40 usd). And you could be the one who wins it.

How? Well, entry is simple, and you can enter in a bunch of ways.

1. Subscribe to email updates of this blog (the subscription form is in the header). – 1 entries
2. Comment on this post, stating what book you want to win. – 2 entry
3. Write a tweet about this contest, and a link back here – 3 entries
4. Blog about this contest on your blog, and a link back here. – 4 entries

The winning entry will be chosen at random. You can mention the book you want in your contest entry. If you don’t, it’s fine, you can still win – I’ll just email you to get the book title after I pick the winner.

Looking forward to seeing your entries, and hopefully it’ll help me find out about some new books as well.

This contest will run until 25 March, 2359 EST.

The contest is now over. And the winner is….KL. Congratulations. I’ll be emailing you shortly.

Quick recap. I had AQ offsuit, raised from the SB, BB 3-bet, and UTG limp-called. I was getting $15.50 to call into a $48.50 pot. In retrospect, even the ranges I put them on were probably wrong. I stand by the range of 77+, AT+, QJ+ for the BB. But considering the UTG limped, I don’t think he had a high pocket pair. So I’m thinking he’s likely to have 77-JJ, QJ+.

So let’s assume I called, and take look at the various cases of the flop. If I called, the pot on the flop would be $64. Obviously I can’t consider every possibility, but let’s look at the main ones.

Case 1: A flops, with no K
The flop contains an A and no K. If this happens, I’m probably quite confident. This happens (3*43*42*3)/(50*49*48) = 13.82% of the time. I’m only afraid of a set if this happens. I’d be first to act, and I’d bet about $40, for a total investment of $55.50. I think JJ, QQ, KK, AT, AJ would at least call that bet. The other hands in the range are probably going to fold. Let’s assume that all the hands in the ranges are equally likely.

The player in the BB will then call when behind about 30% of the time. 11% of the time I’d be behind. 52% of the time, he’s going to fold, and the other 7% is when he has AQ as well and we tie. UTG will call when behind 18% the time, will be ahead of me 10% of the time, fold 64% of the time, and tie the last 8%.

So, betting action. There are a few cases.

1. BB calls and UTG folds, and I’m ahead. This happens 30%  * 64% = 19.2% of the time. At this point, I win $64 + $40 + $40 = $144. And I’ve invested $55.50, giving me a profit of $88.50.
2. BB calls and UTG folds, and I’m behind. This happens 11% * 64% = 7.0% of the time. I win nothing, and lose the $55.50 I’ve invested.
3. BB folds and UTG calls, and I’m ahead. 52% * 18% = 9.4% of the time. I profit $88.50.
4. BB folds and UTG calls, I’m behind. 52% * 10% = 5.2%. I lose $55.50
5. Both call, and I beat both. 30% * 18% = 5.4%. I win $64 + $40 + $40 + $40 = $184, for a profit of $122.50.
6. Both call, and I lose to at least one. 11% * 18% + 30% * 10% + 11% * 10% = 6.1%. I lose $55.50
7. Both fold. 64% * 52% =33.3%. I win $104, for a profit of $48.50.

For now, let’s ignore the cases when I tie with one.

So basically, if an A flops, and no K, I have an EV of 0.192 * 88.5 + 0.70 * -55.5 + 0.94 * 88.5 + 5.2 * -55.5 + 5.4 * 122.5 + 6.1 * -55.5 + 33.3 * 48.5 = $37.86.

Case 2: Q flops, with no A or K.
This happens (3 * 42 * 41 * 3) / (50 * 49 * 48) = 13.17% of the time. This is similar to the above, except I’m behind to KK and QQ as well, and I’m probably being called by QJ/QK, instead of AJ/AK. KK, QQ, AA are not hands I put in UTG’s range (he would have open-raised with those hands, I think, and not limped/called), so I’m not behind to UTG for sure.

So, again, the same few cases of bets.

1. BB calls and UTG folds, and I’m ahead. This happens 36%  * 51% = 18.4% of the time. At this point, I win $64 + $40 + $40 = $144. And I’ve invested $55.50, giving me a profit of $88.50.
2. BB calls and UTG folds, and I’m behind. This happens 9% * 51% = 4.6% of the time. I win nothing, and lose the $55.50 I’ve invested.
3. BB folds and UTG calls, and I’m ahead. 47% * 42% = 19.7% of the time. I profit $88.50.
4. Both call, and I beat both. 36% * 42% = 15.1%. I win $64 + $40 + $40 + $40 = $184, for a profit of $122.50.
5. Both call, and I lose to at least BB. 9% * 50% = 4.5%. I lose $55.50
6. Both fold. 47% * 51% = 24.0%. I win $104, for a profit of $48.50.

This gives me an EV of 0.184 * 88.5 + 0.046 * -55.50 + 0.197 * 88.5 + 15.1% * 122.5  + 4.5% * -55.5 + 0.24 * 48.5 = $58.81.

Case 3: I miss the flop.
When no A or no Q hits. This happens close to 70% of the time, I think. Let’s take 70%, in general. When this happens, I’m probably going to get bet into, and fold, losing the $15.50 I called preflop.

So therefore, my total EV = 0.7 * -15.50 + 0.1317 * 58.81 + 0.1382 * 37.86 = $2.12.

Again, however, it seems like I should make the call here.

But let me place a disclaimer here. The actual EV would be lower if the ranges my opponents are on are tighter. A lot of the calculations made here are based on my read of what hands they’d call a bet with, etc. I understand that the ranges are probably a bit wide, but I feel like the game was quite loose, and those reads are what I genuinely felt.

What do you think, though? Am I thinking through this correctly – or am I missing something else again?

I was playing in a live cash game a few days ago, and was put in what i felt was a difficult decision pre-flop. So I thought I’d do some simple analysis to see if I made the right play.

Here’s how the hand went. It’s a 10-man game, $0.50/$1 blinds. I’m in the small blind, with AQ offsuit. There are a whole bunch of limpers, and I raise to $5.50, in an attempt to narrow the field, figuring some of the limpers are weak and would fold.

Firstly, was this raise right? There were about at least 4 limpers. And even against 4 more random hands, AQ is only about 30% favorite, and I didn’t want that risk. I wanted to narrow it down, and that was why I made the raise. My equity increases exponentially the more players fold, and it gives me more information.

The big blind, a moderate-tight player, who does bluff at times, but not very often, re-raises to $21. UTG, who has been quite tight passive so far, calls, and the rest all fold to me. After some consideration, I folded. I was basically thinking that, I’m either against a lower pocket pair, where I’m in a coin toss, a hand similar to mine, or a higher pocket pair, where I’m crused. So I figured I was either 50-50 or crushed, and so I folded. But was this the right decision?

Let’s take a look at it. Right now, the pot is about $48.50. I’m left with $15.50 to call.

That means that for me to call, I need to have an equity x of

EV > 0
x * (15.50+48.50) – 15.50 > 0
x * (64) > 15.50
x > 15.50/64 = 24.22%

What kind of ranges would I have a 24% equity against? If I put both players on a range 99+, KQ+, that gives me an equity of about 23%. At 23.2%, a case can be made for calling, because of the implied odds of the flop (if I hit an A against KK, for example, I’m bound to win a bit more). So even if both players are on 99+, KQ+, I think I can call here and see a flop.

Obviously, however, the two players would have slightly different ranges. I would think the big blind is on a range like 77+, AT+, QJ+, and the UTG player on a stronger range of 99+, AQ+. Even then, I still have 24.25% equity. Again, that’s not counting the implied odds, the odds of the big blind bluffing, etc.

So, it seems like the fold was a mistake, based on the numbers. What do you think? Would you have folded?

Poker Stars $0.01/$0.02 No Limit Hold’em – 8 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

CO: $1.92
BTN: $2.93
SB: $1.11
Hero (BB): $2.58
UTG: $3.00
UTG+1: $4.83
MP1: $2.64
MP2: $3.33

Pre Flop: ($0.03) Hero is BB with Q J
UTG calls $0.02, 3 folds, CO calls $0.02, 1 fold, SB calls $0.01, Hero raises to $0.10, 2 folds, SB calls $0.08

Flop: ($0.24) 2 2 8 (4 players)
SB checks, Hero checks

Ignore the preflop bet sizing, I know there’s already beena lot of debate over that. But there were a lot of people saying that I should have bet the flop, to get value from my draw. Basically the premise is that I won’t get paid off much if I hit the draw. Sounds logical, but let’s try and see quantitatively how true that is, and how much that bet should be.

Let’s assume that my opponent would fold if I hit the flush. On this flop, I have 9 outs to the flush, giving me a 9/47 = 19.1% chance of turning a flush, and 35.0% of hitting a flush by the river. However, we’ll take the odds of the flush hitting on the turn, because to take the case of the flush coming on the river, we have to consider if the opponent will bet a blank on the turn. Maybe I’ll consider that in a later post.

So let’s say a diamond comes on the turn, and my opponent shuts down here. If I had bet x cents on the flop, that gives me an EV of 0.191 * (24 + 2x) – x = 4.596 + 0.383x – x = 4.596 – 0.617x. For it to be a good play, that gives

EV > 0
4.596 – 0.617x > 0
4.596 > 0.617x
x < 4.596 / 0.617
x < 7.45

So I’d have to bet about 7c for it to be a good play. That’s a bit too small a bet, though, to make. If I bet that, he’s likely to raise (I would if I was the SB facing a 7c bet into a 24c pot), so that kind of doesn’t make sense. If I had to bet the flop, I’d have to bet something like 14c, at least. Let’s take a bet of 14c, that gives me an EV of 0.191 * (24 + 28) – 14 =-4.04. So it’s slightly negative.

I do have some sort of fold equity here. How much fold equity do I need to make this a positive play? Assuming fold equity f, that gives

EV = f * 24 + (1-f) * -4.04 > 0
24f – 4.04 + 4.04f > 0
28.04f > 4.04
f > 4.04 / 28.04
f > 0.144

Therefore, I need him to fold at least 14.4% of the time for that to be positive. I think that’s definitely a reasonable assumption, so a bet of 14 cents seems like it’s ok.

The question then becomes, what’s the relationship between the bet size, and the fold equity required to make it a positive EV play?

For the play to be +EV, you need the following to be satisfied, taking f as fold equity and b as the bet size.

EV > 0
f * 24 + (1 – f) * [0.191 * (24 + 2 b) - b] > 0
24 f + (1 – f) * (4.596 – 0.617 b) > 0
24 f + 4.596 – 0.617 b – 4.596 f + 0.617 * b * f > 0
19.404 f -0.617 b + 4.596 + 0.617 * b * f > 0
f (19.404 + 0.617 b) – 0.617 b + 4.596 > 0
f (19.404 + 0.617 b) > 0.617 b – 4.596
f > (0.617 b – 4.596) / (19.404 + 0.617 b)

I don’t think you’re really ever betting anything less than half the pot (12 cents) here – I know I wouldn’t. So, for bets of 12c to 24c, here are the required fold equities to make this a +EV play.

Let’s say your opponent is folding 30% of the time to your bet (which I think is a reasonable percentage). The more you bet, the lower the expected value is. The EVs for various bet sizes, assuming 30% fold equity, are as follows

Your EV from checking this street is simply 0.191 * 24 = 4.596. To make the bet better than checking, you require an EV of more than 4.596, which suggests a bet size of 12 or 13 cents. I’d probably include 14 (and maybe even 15 cents) in that range, because you will get paid off after hitting the flush every once in a while, so the implied odds would increase the EV. That depends on a lot of variables, of course – who the opponent is, your table image, stack sizes, etc.

But based on the analysis so far, I’d say that a bet of 12 cents to 15 cents, or half to about 60% of the pot, seems to be the best way to get value from a flush draw. Anything more is a bit too much, considering you’re only on a draw, and anything less is probably a bit too weak and likely to get raised.

What do you think? Do you bet your flush draws on the flop for value, in case you don’t get paid when the flush hits? If so, how much would you bet your draws?

So, today I want to take an initial look at this. Say, you’re dealt a hand like KK. You raise preflop, there are some folds, 1 player re-raises, it folds around back to you, and you 4-bet preflop. For each possible preflop hand, what are the odds that he will call/raise your 4-bet?

Let’s assume that you don’t have much data on the player, and you’re looking at hand strength alone. And also assume that your opponent is a decent player. For certain hands, it’s easy to predict. With AA he’s always at least calling the 4-bet. For a hand like 7-2, there’s practically no chance that he’ll call a 4-bet.

What about in the middle, though? After a discussion with a friend of mine, we concluded that the distribution would look something like this:

Basically, the horizontal axis shows represents the strength of the hand (with the weakest hand on the left, and the strongest on the right). The vertical axis is the probability of calling (or raising) a 4-bet. Essentially, what the graph means is this. There are a number of weak hands, which players are highly unlikely to call with. 7-2, 9-3; these are all hands which are almost certain to be folded. There are also certain hands which are players will almost definitely call with: AA, KK, QQ, etc. That’s why the extreme left and extreme right side are more or less flat.

In the middle, though, are the marginal hands. With these hands, players might sometimes call, depending on the player’s ability, the situation at the table, a number of factors. But essentially, we believe that the marginal hands would have an increase in frequency of calling, based on the strength of the hand. Basically, there is almost no difference between how often a player would call with AA and KK, or how often he’ll fold 7-2 or 9-4. But there will be larger differences between how he plays KQ offsuit, and 99, for example.

So based on that hypothesis, I’ve created a chart to predict a probability of how often a player will not fold (ie call/raise a 4-bet). I used the Chen formula to determine hand strength, and grouped the hands roughly. Based on the hand strength and the groups, I then assigned a probability, with the lowest group distributed over 0 to 5%, the highest group over 95 to 100%, and the middle groups distributed across the remaining range (with a few more subtleties, but that was the general idea. What I got was this.

To be honest, it doesn’t quite match what I want. The problem with the Chen ratings is that the hands tend to be bunched up together. I’ll look into other ways to evaluate hand strength, but so far, this is my initial (and admittedly very crude model). I’m thinking for the next step, I’ll use the WestonPoker PreFlop Odds chart instead of the Chen formula as a basis for hand strength. I think that will give a much smoother curve. I’ll do that and post an update soon.

What do you think, though? How do you think the chart of a 4-bet call/raise would look like? Am I on the right track or what would you have done differently?

I’m still not fully sure how I should be playing early tournament, to be honest. I’ve mainly been trying to play tight, and not risk too much. But because of that I’ve never really been able to build too much of a stack. So that’s enabled me to last past the first break, but by the time it’s around the second break, my stack isn’t really large enough to maneuver with, and I end up having to shove.

I think both times I’ve busted out, it wasn’t really a misplay on the hand that busted out. In the first one, I had about 15 big blinds, was dealt AQ. Large stack raised preflop, I re-raised and shoved. The large stack called, and showed QQ, which held up. I got it in bad, but I think at 15 big blinds, and with antes, I really had to make some sort of move.

Event 4 was just a really bad beat. I had pocket Aces, with just over 10 big blinds. The flop came K T 8, I got it all in, and my opponent showed JT. The turn came a T to knock me out.

So, I don’t think I mis-played either of these hands. But my main concern is that I’m getting in these spots in the first place. I don’t seem to be able to build a large enough stack, and because of that, once antes come in, I’m in a dangerous position and have to shove more often than I’d like. But I’m also worried that if I play too aggressively, I might bust out really early.

What do you think? What’s your strategy for the early stages of a tournament? Do you go aggressive, to try and double up early (and risk busting out early), or do you try to maintain your stack and grind it up stack slowly?

Blinds are 100/200, John has A J and is using the ‘Small Ball‘. He minimum raises from a latter position and Dave calls it. Flop is 7 6 J, good flop for John, Dave checks and John makes just a weak bet of 250 in a 900 pot. Seeing this Dave calls.

So, John hits top pair, top kicker on the flop. There are two connected cards, and the pot is 900. He bets 250 into the 900, giving Dave odds of about 27.8%. For the sake of argument, let’s assume it’s a rainbow flop.

Is this the right bet size? Remember, the main question here is whether the small ball strategy works, or whether it is (as TrueGamble says), “A New Way of Trapping Yourself”.

What could Dave have in this situation? Given the preflop bet of 2 big blinds, Dave could probably have any two cards, any of the 1081 total possible hands. Assuming that, what hands does Dave have that could beat John? At this point in time, John is only losing to a 6-6, 7-7, J-J, Q-Q, K-K, A-A, J-7, J-6, 7-6, a total of 30 hands. There are 6 hands (other A-J combinations) that would tie with John. Assuming all the hands for Dave are equally likely (we assumed that Dave could have any two cards), that gives John’s hand a current strength of 96.85% (ie. if play were stopped at this point in time, John has a 96.85% equity). Even when he’s behind, though, he has about a 11.5% chance of improving to win (if Dave has 6-7, for example, any A or J would give John the winning hand again).

Given that, at this point in time, John has a pretty strong hand, and should definitely bet. The question is then, how much. When Dave is behind, at this point in time, he has at most 8 outs. Let’s assume he has 9-T (the best hand he could have that’s behind). He then has 8 outs, and a 17.8% chance of hitting one of those 8 outs on the turn. If John bets 250 into the 900 pot (as in the example), Dave’s call would yield an EV of 0.178 * (250+900+250) – 250 = -1.11, which are still the wrong odds to call.

But Dave isn’t necessarily sure that John has the J. If John has the 7 or 6, or no pair, Dave has 6 additional outs. If we use 14 outs, then (the straight draw + the draw to any pair), then Dave has about 31.1% of hitting an out, and an EV of 0.311 * 1400 – 250 = 185.56, making it a profitable call. So if Dave believes John is essentially bluffing without TP, he has an EV of 185.56, and if he believes John has TP, he has an EV of -0.8.  That means his overall EV is -0.8x + 185.56 (1-x) = -1.11x + 185.56 – 185.56x = -186.36x + 185.56, with x = probability that John has TP.

To make this profitable, then,

EV > 0
185.56 > 186.36x
x < 185.56 / 186.3 = 0.994

Essentially, this means that unless Dave is very sure that John has the TP, he should be making the call. I don’t think that in this situation, it is ever possible to say about 99.4% certainty that John has TP, so with a hand like 9-T, Dave definitely has to call.

Now, let’s say John bets 450 instead. Using the same calculations, John needs to have TP at least 45.8% of the time for Dave to make the call. This is a much tougher spot, putting Dave in more or less a 50-50 position. If John’s bluffing about half the time, then Dave should not make this call. If the bet is 600, that bluffing percentage then has to be below 19%.

All this is assuming Dave has a hand like 9-T, which is the best case scenario, so we might want to lower the percentages slightly to account for the other hands.

What does this all lead to? This means that Dave is almost definitely going to be right in calling a bet of 250, no matter what hand he has. So it would seem that 250 is definitely too small a bet. If John bets between 450 to 600, however, Dave would need to believe that John is bluffing at least 55%-80% of the time, based on the bet. That makes it a much more marginal call. I think that it’s going to be hard for Dave to say with anything more than 60-70% certainty that John doesn’t have the J, especially with the higher bet sizes.

So, my conclusion from this analysis? I would believe that a bet in the range of 450 to 600 is a better bet, somewhere between half to 2/3 of the pot.

What do you think? How much do you generally continuation bet on the flop, and why?