By Request, an Intro to Poker Math

[jaxUp]

we'll start with a couple definitions

Probability: the probability of event A occuring is:

number of ways A can occur
-----------------------------
number of possible outcomes

so, suppose A is the event of catching a 7 on the river, when we hold 98o, and the board is A856. The probability of a 7 hitting is:

4/46 (since 4 possible 7's, and 46 unknown cards)

It is often expressed as a decimal or percentage.

- the sum of the probabilities of all possible outcomes of a situation is 1
- 0 <= probability of some event <= 1



Odds Very directly related to probability.Particularly in poker we tend to talk about the odds against something happening. Suppose we know the probability of event B is x/y then the odds against B occurring are
(y-x):x

so, from our example above, the odds against rivering a 7 would be (46-4):4 = 42:4 = 10.5:1.

We use this when trying to figure out pot odds. Specifically in this case we would know that there have to be 10.5 BB in the pot for the call to be at least neutral EV.

Multiplication and Addition Rules

just remember this:

multiplication = and
addition = or

the probability of hitting a BDFD (hitting a flush card on the turn AND river) is:

Probability turn is a flush card * probability river is a flush card

= 10/47 * 9/46 ~ 0.042 = 4.2%

(note that we assumed for the river that the turn was a flush card. If people want to get nitpicky, I know that I should really say that the first part is multiplied by the probability that the river is a flush card GIVEN the turn is also a flush card. However, to those nits, I say [censored] you. If you did not understand this, congratulations, you are not a nit, and you can move on (ignoring this part))

Now, if we flop a FD, the probability of making it by the river (ie hitting a flush card on the turn OR the river) is:

probability turn is a flush card + probability river is a flush card

= 9/47 + 9/46 ~ 0.3871 = 38.71%

(again, [censored] you to all the nits)

************************************************** ******

OKAY, believe it or not, that's really all you need to know. Now you just need to be able to put it together for EV calculations. If there are other basic questions you have I (or somebody else) will be happy to answer them I'm sure. Don't worry about sounding like a retard. It's the internet, and nobody cares.

Basically, to find the EV of a particular line, you have to take into account all the possibilities that will happen on each street. Here's a simple example, where we decide whether or not bet the turn with air.

2 limpers, hero raises A[img]/images/graemlins/club.gif[/img]J[img]/images/graemlins/club.gif[/img] OTB, BB calls, and limpers call.

Flop (8SB): 2[img]/images/graemlins/spade.gif[/img]3[img]/images/graemlins/diamond.gif[/img]9[img]/images/graemlins/club.gif[/img]
check, check, check, Hero bets, fold, fold, call

turn (5BB): 2[img]/images/graemlins/heart.gif[/img]
check, hero???

ok, suppose that based on villain's range we are ahead here 35% of the time (and villain will fold in these cases). So, we are behind 65% of the time. In these cases villain will call a turn and river bet. Let's say that we have 5 outs on average when behind. Also, if we choose to check behind on the turn, villain will bet the river every time. If we raise him he will only call with a pair. Also, assume he has 5 outs when behind

so let's compare betting the turn and checking behind on the river UI with checking the turn and calling a river bet.

BETTING THE TURN:

we will hit the river and be good 5/46 ~ 10.87% of the time
so we will lose 1BB .65 * .8913 (when we are behind AND miss the river)
we will win 7BB .65 * .1087 (when we are behind AND hit the river)
we will win 5BB 35% of the time

so the EV of betting the turn is 5*.35 + 7*.65*.1087 - 1*65*.8913

= 1.67 BB

NOW LET's COMPARE IT TO CHECKING BEHIND AND CALLING a BET

we lose 1BB (.65 * .8913) + (.35*.1087) (we are behind AND miss OR we are ahead AND he hits
we win 6BB .35 * .8913 (we are ahead and he misses)
we win 7BB .65 * .1087 (we are behind and we hit)

so, the EV of this line is ~ 1.75 BB


So, comparing the 2 lines, checking behind and calling a bet is marginally better, although not a huge difference. But anyways, that's how you calculate the EV of a given line. Consider all the possibilities, and how much you will lose in each one, and then add them up. When trying to figure out if you've covered all the possibilities, it may be helpful to remember that the sum of the probabilities for all events is 1. **so you'll note from when I examined the second line that (.65*.8913 + .35*.1087 + .65*.1087 + .35*.8913 = 1).

If anybody is feeling real adventureous they can give the following hand a try. If you want, just try doing the EV calculation for 1 particular line. I made a whole bunch assumptions that should be enough to do it, but feel free to make any more non-trivializing assumptions you see fit.


Ok, here's a hand I played earlier today

Party Poker .5/BB/1BB

Hero is dealt T[img]/images/graemlins/club.gif[/img]J[img]/images/graemlins/diamond.gif[/img] in CO

3 folds, Hero raises, 2 folds, BB 3bets, Hero calls

flop (6.25 SB): 2[img]/images/graemlins/diamond.gif[/img]6[img]/images/graemlins/club.gif[/img]J[img]/images/graemlins/spade.gif[/img]

BB bets, hero......???????

OK, suppose I have the following range for BB, based on my stellar reads:
88-AA, ATs+, AJo+, KQs

My stellar reads also tell me a couple other things. Villain will cbet the flop and turn here every time. If he has a pair, but less than TPTK, he will call down vs a raise. If he has TPTK+ he will 3bet. If we raise the flop and he has no pair, he will call and c/f the turn UI. If we call the flop and raise the turn and he has no pair and no draws he will fold. If we call the flop and turn, and bet the river when checked to he will call with AQ UI+ So, what line do we want to take? look at the following possibilities:

1) raise the flop and go from there
2) call flop, raise turn and go from there
3) call flop, call turn, put in 1 bet on the river

[Commie]

Hey man great post, some useful stuff here!

I did have a question about probability though. I was trying to figure out the probability of hitting at least one of your cards on the flop. Let's say you hold A[img]/images/graemlins/heart.gif[/img]K[img]/images/graemlins/spade.gif[/img] and want to know how often an ace or a king will hit the flop.

I think (forgive me if this math is wrong, I sux at probability) you can do (44/50)*(43/49)*(42/48) to get the probability of not hitting your card and subtract it from 1, to get 32%.

But is there another way of doing it without doing 1 - (...) ???

[Str8Fish]

There are a couple ways to do it other than 1-(...). But seriously you don't want to go that direction. It involves Combinations and Permutations and it's all not that pretty. If you're still interested to learn a harder method I could tell you, but it's all so much easier just subtracting it from 1.

EDIT to add, I think since you're the one who asked for this post, that you should try to solve the question at the end of Jax's post. Let's see if you can apply what you've read.

[Commie]

Lol, I didn't ask for this post O.o

I'll try to solve it anyway thank u very much (in my head)!

But thanks for your reply. Does the harder way involve using the "choose" stuff.

[jaxUp]

[ QUOTE ]
Does the harder way involve using the "choose" stuff.

[/ QUOTE ]

The harder way involves breaking it down into a case by case basis...It's complex enough that it's not really worth considering when it's so easy to do it the 1-x way.

[Str8Fish]

oh right manhammer asked for it. Where's he at with his proof?

and if you can do the math in your head, you're my new math god.

[MrWookie]

Great Post, jax. For those of you looking for some examples of this in action, check out the most recent digest and take another look at Str8Fish's work. I really kicked his ass on some of these topics, and he came through big with a definitive calculation.

[Befolder]

Nice post Jax.

The only thing I will struggle with is how to make these EV calculations of each choice quickly at the table.

Currently, it's still too complicated for me to figure out how I would have time to apply it during a hand. If I could, I'm sure it would improve my decision making quite a bit. I've been winning for three years now and never busted out, but there's nothing wrong with winning more, which is what I'd like. I just may have to do it by continuing to work on my postflop game.

I do give my oppenent possible hands and make my decisions based on that, but I'm sure I'm making marginal EV decision mistakes, raising when I should call. Checking when I should bet. Whatever.

Again, nice job.

[Sushiglutton]

[ QUOTE ]
Now, if we flop a FD, the probability of making it by the river (ie hitting a flush card on the turn OR the river) is:

probability turn is a flush card + probability river is a flush card

= 9/47 + 9/46 ~ 0.3871 = 38.71%



[/ QUOTE ]

This is not exactly correct. If we have a flushdraw on the flop the chance to make a flush (one or both turn and river are flushcards (fc)) is:

(9/47)(38/46) + (38/47)(9/46) + (9/47)(8/46) = 1-(38/47)(37/46) = 35.0%

'The chance that the turn is a fc while the river is not' + 'The chance that the river is a fc while the turn is not' + 'The chance that both the turn and the river are fc' = 1 -'The chance that neither turn nor river are fc'


The chance that turn or(strict or) river are fc:
(9/47)(38/46) + (38/47)(9/46) = 31.6%


The reason ur calculation is incorrect is that the two events:
A = 'The turn is a flushdraw'
and
B = 'the river is a flushdraw'

are not disjunkt (in swedish [img]/images/graemlins/wink.gif[/img] ) (meaning: A AND B != 0 ('!='-not equal to) that is: it's possible that BOTH A and B are true) and thus: P(A U B) != P(A) + P(B)

[Str8Fish]

[ QUOTE ]
(again, [censored] you to all the nits)

[/ QUOTE ]

Maybe you didn't see this part of his post.