A bit of smart poker math

Great article about mathematics of poker from OlegK.

I have recovered from a short vacation and found that some thoughts of my students can sometimes cause either laughter, or frustration. I really want to discuss one issue, that has plagued almost all, without exception, lower limit players: “How much should I bet?”.

Really, how much? I imagine a pile of questions raised in your head now:

• What if he will get his flush on turn and I will lose a lot?
• What if he will get his flush on river?
• I’m in my downswing and they call me all the time anyway…
• I have a nuts, so I will be betting less on this draw-heavy board and let him call with his weak draw…
• I have a weak hand - I don’t want to bet large…

So fellas, now I will try to methodically input some simple numbers into your head, that will show you what and how you should bet.

Before I’ll start, I will tell you that for the first time I was talking this information with my friend who was betting half-pot continuation bets during his downswing, arguing that everyone calls his bets and he is check/folding on turn – that’s why he will lose less. I guess I even have a video evidence of this shame. :)

Well, now jokes aside (which are not completely jokes anyway). What you need to be thinking of before placing a bet? Definitely about our expected value (EV). And after that we can give answers to some questions based on the logical exclusion of options. But math first.

So let’s first prove that our continuation bet with many-many hands on a huge amount of flops in vacuum will be profitable. What we need for that: PokerStove and a pair of hands. Nothing else. Oh, and an EV from bet formula. Let me be clear, this formula will provides only with expected value of our bet on flop. In real game situation this will be different, but we will have a very good understanding of how profitable our continuation bet is, because it’s quite difficult to play turn and river with losses without significant issues in your game.

Let’s start with simple things. Imagine that we have a top pair with a good kicker (I’m not even talking about something better) on a flop with an average amount of draws. For example,     on a       board.

That’s set. Now let’s decide with how many cards our opponent will call in this situation. I must say, that we will not take into consideration all the scenarios where our opponent will fold his hand and we will win the pot instantly. So we’re taking the worst-case scenario, where our opponent in his range has only the hands he will call with. It’s obvious, that this scenario will be just awful for the cases when we have complete bluffs, because in that case they lose all their reasoning (why you would bluff if you will always get called?). On top of that we will no take in consideration the scenarios where we will get check/raised on flop. They are also looking not appropriate, considering that we’re not discussing the situation where an opponent just folds.

Let’s decide the call range of our opponent now. Let’s say it will be: KJ, KQ, 89 (only one combo), JT, QT (we are thinking that other flush draws he will play with check/raise line), T9, T9 (other combination he will check/raise), 65, 65 (other combination he will check/raise), 99, TT. Here’s just an average range. Now let’s launch PokerStove (or HoldEq) and start to count our equity:

Board:      
Hand 0: 73.693% { KhQd }
Hand 1: 26.307% { TT-99, KJs+, QsTs, JsTs, Tc9c, Th9h, 9c8c, 6c5c, 6h5h, KJo+ }

It appears, that we are strongly ahead of that range, which is not surprising. Now let’s calculate the EV of our bet:

EV (of bet) = Our equity * (Pot size after opponent calls) - Our investment

Suppose we put 2/3 of pot - $10 into the pot of $15, so we have:

EV (of bet) = 73,6% * ($15+$10+$10) - $10
EV (of bet) = +$15,76

Let’s not consider different nuances of bet balancing, or tendencies to call or fold depending on the size of our continuation bet. Otherwise I would need to write a few books and that will not be free.

In that situation it’s obvious, that the larger is our bet, the better it is for us, because we’re ahead of opponents range. And the bigger the price of his fault (call against the odds), the better it is for us. So if we will bet a full pot and get a call from the same range (which is completely realistic), our EV will be in the area of $18.

Let’s keep going, no we will methodically decrease the strength of our hand. Now we’re talking about the flop only. We will discuss what will happen on turn later on.

Let’s say, that we have JJ on the same flop. What has changed?

Board:      
Hand 0: 45.113% { JdJh }
Hand 1: 54.887% { TT-99, KJs+, QsTs, JsTs, Tc9c, Th9h, 9c8c, 6c5c, 6h5h, KJo+ }

Now against the same range we’re a little weaker than 50/50, however...

EV (of bet) = 45,1% * ($15+$10+$10) - $10
EV (of bet) = +$5,78

We’re still playing profitable poker! And our EV doesn’t change significantly from our continuation bet size. That is, with our second pair we still can’t play check or bet less, even when our opponent never folds anything.

When things will go worse? Let’s take 89 in our hands. Now:

Board:      
Hand 0: 25.776% { 9h8h }
Hand 1: 74.224% { TT-99, KJs+, QsTs, JsTs, Tc9c, Th9h, 9c8c, 6c5c, 6h5h, KJo+ }

We’re finally losing money! But how much?

EV (of bet) = 25,7% * ($15+$10+$10) - $10
EV (of bet) = -$1

In this situation the more we put into pot, the more we will lose, which should be obvious - we’re not favorites in the hand anymore, and the more we will invest into pot with the hand weaker than our opponents, the worse it is for us.

Did we found the limit where we should be betting 1/2 of the pot? Nope. We will still be betting 2/3, first of all to balance the range of our flop betting (against someone who can’t be exploited that way - for example fish, that will call you with all possible draw and ace-high, and will never play check/raise), because we will also be getting folds from hands like AJo, ATo, etc. And the amount of such hands will be much larger, than the ones with which our opponent will decide to play check/raise (sets, two pairs, and very strong draws – there are not more than 12-14 of such combinations, when there are 12 combinations of only AJo). Thus, any our continuation bet with any pair will be quite profitable, considering our chances to get two pairs or trips on turn for example. Besides, quite often when betting 1/2 of the pot on flop we will win about $1 of EV in vacuum, but you will get a lot more of check/raises exactly because of the size of your bet. Also, $1 in a game where the pot on flop is $15 already – that’s about a half of big blind. And that’s only in the most dreadful case for your weak pair. Same logics can be used for the other 55 pocket pairs as well.

Now let’s see if we should bet less with our draws and complete air?

Let’s say we have    .

That is just two simple overcards on flop, which can catch some draw or pair on turn to be able to bet second barrel. Thus, our equity is:

Board:      
Hand 0: 22.928% { AdJh }
Hand 1: 77.072% { TT-99, KJs+, QsTs, JsTs, Tc9c, Th9h, 9c8c, 6c5c, 6h5h, KJo+ }

Not that bad, isn’t it? We’re standing not much different to the situation where we have a pair of eights:

EV (of bet) = 22,9% * ($15+$10+$10) - $10
EV (of bet) = -$1,98

And again, we do not consider situations where we will win pot on the flop. Besides, 2/3 of the pot is not a bad size.

What next? Let’s give us a draw, for example QT (not the nuts flush-draw and not a monster draw).

Board:      
Hand 0: 46.449% { QsTs }
Hand 1: 53.551% { TT-99, KJs+, QsTs, JsTs, Tc9c, Th9h, 9c8c, 6c5c, 6h5h, KJo+ }

I think you got it...

Thus, we now understand that, in principle, continuation bet on flop in a size of 2/3 of pot is oftentimes quite profitable. You shouldn’t abandon it for quite obvious reasons – with the most of our hands this bet will be profitable even against the range that will never fold to continuation bet, but with other hands we will just hope for a fold from opponent, and our bet will not unprofitable on its own. On top of that it’s easily balanced either with bluffs or value hands – we could try to find a balancing point, but based on EV distribution charts from bets with different hands, 2/3 of the pot is a very suitable sizing. In general, we will often play with quite an unexploitable style on flop against an unknown opponent.