By Request, an Intro to Poker Math
 

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

Re: By Request, an Intro to Poker Math
 

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 - (...) ???

Re: By Request, an Intro to Poker Math
 

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.

Re: By Request, an Intro to Poker Math
 

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.

Re: By Request, an Intro to Poker Math
 

[ 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.

Re: By Request, an Intro to Poker Math
 

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.

Re: By Request, an Intro to Poker Math
 

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.

Re: By Request, an Intro to Poker Math
 

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.

Re: By Request, an Intro to Poker Math
 

[ 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)

Re: By Request, an Intro to Poker Math
 

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

[/ QUOTE ]

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

Re: By Request, an Intro to Poker Math
 

Oooooh, sorry [img]/images/graemlins/blush.gif[/img] [img]/images/graemlins/blush.gif[/img] [img]/images/graemlins/blush.gif[/img] [img]/images/graemlins/blush.gif[/img] [img]/images/graemlins/blush.gif[/img] [img]/images/graemlins/blush.gif[/img]

Re: By Request, an Intro to Poker Math
 

Edited because Sushi beat me to it.

And, Str8, if you're going to do something you should do it right, dontcha think? 38% vs. 35% can be the difference between this being a value bet/raise on the flop and not, dontcha think?

Re: By Request, an Intro to Poker Math
 

[ QUOTE ]
[ 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)

[/ QUOTE ]

I think doing all that extra math is -EV, and worth whatever time it took times your win rate in trade. Somebody would have to do the math to figure out how -EV doing that extra math is, but again that would also be -EV.
[img]/images/graemlins/cool.gif[/img]

Re: By Request, an Intro to Poker Math
 

<font color="red">* * * NIT ALERT * * *</font>

Re: By Request, an Intro to Poker Math
 

[ QUOTE ]
<font color="red">* * * NIT ALERT * * *</font>

[/ QUOTE ]

Tough to be a nit when I don't even follow half of the complicated math stuff, so me seriously making a complaint about it or opposing it would be fruitless.

I was just making a joke.
[img]/images/graemlins/smile.gif[/img]

Re: By Request, an Intro to Poker Math
 

sushi, right on with your correction. I double-counted the situations where a flush card hits on the turn and river.

I made a note of it, but for some reason did not think it would make much of a difference in the calculations. I was certainly wrong in this regard. 3% is nothing to scoff at, and I stand humbly corrected.

(oh, and disjunkt is mutually exclusive, or distinct in english)

Re: By Request, an Intro to Poker Math
 

befolder, as you can see, the EV calculations can get a little complicated and you really can't figure them out at the table a lot of the time. However, when you review your hands you can do them then, and when the same situation arises next time you will already know how to maximize your EV.

Re: By Request, an Intro to Poker Math
 

Guess nobody else is doing the hand example at the end, so give me 30 minutes and I'll have it pumped out for you all.

Re: By Request, an Intro to Poker Math
 

SWEEEEET!!
Thanks a lot, jax. I'm sure this will be very helpful. Unfortunatly, I have to leave for work [img]/images/graemlins/frown.gif[/img], so I'll have to wait until tonight to really get into it. Thanks again.

Re: By Request, an Intro to Poker Math
 

[ QUOTE ]
Guess nobody else is doing the hand example at the end, so give me 30 minutes and I'll have it pumped out for you all.

[/ QUOTE ]

str8fish, please make it explicit so it's easy for others who aren't as experienced to understand.

Re: By Request, an Intro to Poker Math
 

[ QUOTE ]
[ QUOTE ]
Guess nobody else is doing the hand example at the end, so give me 30 minutes and I'll have it pumped out for you all.

[/ QUOTE ]

str8fish, please make it explicit so it's easy for others who aren't as experienced to understand.

[/ QUOTE ]

I'm trying as hard as I can to make it so. I'm almost done with the EV of raising. I'll post each separately.

Re: By Request, an Intro to Poker Math
 

PART I- Raise the flop and go from there

Range of hands for villain:
88 – (6 combinations) [8[img]/images/graemlins/diamond.gif[/img]8[img]/images/graemlins/spade.gif[/img], 8[img]/images/graemlins/diamond.gif[/img]8[img]/images/graemlins/heart.gif[/img], 8[img]/images/graemlins/diamond.gif[/img]8[img]/images/graemlins/club.gif[/img], 8[img]/images/graemlins/spade.gif[/img]8[img]/images/graemlins/heart.gif[/img], 8[img]/images/graemlins/spade.gif[/img]8[img]/images/graemlins/club.gif[/img], 8[img]/images/graemlins/club.gif[/img]8[img]/images/graemlins/heart.gif[/img]] – calls down versus a raise
99 – (6 combos) - calls down versus a raise
TT – (3 combos, since hero has a T) – calls down versus a raise
JJ (1 combination – J[img]/images/graemlins/club.gif[/img]J[img]/images/graemlins/heart.gif[/img]) – villain 3-bets
QQ-AA – (18 combos) – 3-bets

ATs (3 combinations… cannot make 4 since hero has T) – villain calls, c/f the turn ui
AJs ( 2 combos with J on flop, J with hero) – 3-bets
AQs (4 combos) – calls and c/f the turn ui
AKs (4 combos) – calls and c/f the turn ui

AJo (4 Aces * 2 Jacks = 8 combos – 2 combos suited = 6 combos) – 3-bets
AQo (4 Aces * 4 Q’s = 16 combos – 4 combos suited = 12 combos) – calls and c/f the turn ui
AKo (16 combos – 4 combos suited = 12 combos) – calls and c/f the turn ui

KQs (4 combos) – calls and c/f the turn ui


Scenarios for a raise on the flop:

1) Villain calls down versus a raise on the flop and does not improve

Hero contributes 3SB preflop. Raising the flop he puts in 2 more. Then he puts in 4 more SB on the turn and river. = 4.5BB contributed to win a pot that is 6.25SB preflop, 10.25SB flop, 14.25SB turn, 16.25SB river = 8.125BB final total pot.

2) Villain calls down versus a raise on the flop and does improve on the turn/river

Hero contributes 3 SB preflop. Raise flop, 2 more SB. BB c/r turn (2 BB). Hero calls, calls river bet (1 BB) [Assuming Hero just calls a river bet even if he himself improves]. Hero contributed 5.5BB to win a pot that is 6.25SB preflop, 10.25SB flop, 18.25SB turn, 22.25SB river = 11.125BB final total pot.

3) Villain calls raise on flop and c/f the turn ui.

Hero contributes 3 SB preflop. Raise flop, 2 more SB. BB c/f the turn ui. We contributed 5SB to win a 6.25SB preflop, 10.25SB flop = 5.125BB pot.

4) Villain 3-bets a raise on the flop and we fold the turn ui.

Hero contributes 3 SB preflop. Raise flop, get 3-bet, 3 more SB. BB bets turn and we fold. = -3BB.

5) Villain 3-bets a raise on the flop and we improve on the turn and call down.

Hero contributes 3 SB preflop. Raises flop, gets 3-bet, 3 more SB. BB bets turn, we call (1 BB), BB bets river, we call (1 BB). Hero contributes 5BB to win a pot that is 6.25SB preflop, 12.25SB flop, 16.25SB turn, 20.25SB river = 10.125BB pot.


EV calculations

Ok, so according to our hand range, we have:

15 combinations that plan to call down versus a raise.
27 combinations that 3-bet a raise on the flop.
36 combinations that intend to call and c/f the turn ui.
Sum = 75 combinations total

EV(raise) =
[villain calls down versus raise]
[88-TT] 15/78 * (8.125BB won * (43/45*42/44) [odds villain doesn’t improve with his 2 outer] – 5.5BB lost * (1 – 43/45*42/44) [odds villain does improve]) +

[villain 3-bets a raise]
[JJ] 1/78 * (-3BB * (42/45)[no T on turn] – 5BB * (3/45)[T on turn]) +

[QQ-AA] 18/78 * (-3BB * (40/45)[no T/J on turn] + 10.125BB * (5/45)[T/J on turn] * (42/44)[no 2 outer on river] – 5BB * (5/45)[T/J on turn] * (2/44)[villain hits 2 outer]) +

[AJo/s] 8/78 * (-3BB * (42/45)[no T on turn] + 10.125BB * (3/45)[T on turn] * (40/44)[no J or A on river] – 5BB * (3/45)[T on turn] * (4/44)[J or A on river]) +

[villain c/f the turn ui, calls down if improves]
[AK,AQ,KQs] 36/78 * (5.125BB * (39/45)[villain hits none of his 6 outs] – 5.5BB * (6/45)[villain hits one of his 6 outs on turn] * (39/44)[no J or T on river] + 11.125BB * (6/45)[Villain improves on turn] * (5/44)[J or T on river]

= [88-TT] +1.33BB + [JJ] -0.04BB + [QQ-AA] -0.37BB + [AJ] -0.23BB + [AQ+,KQ] 1.83BB

EV(raising flop) = +2.52BB


Ok, so continuing in this hand is definitely +EV. Folding in this spot would be a big mistake. Parts II and III will be coming soon.

Re: By Request, an Intro to Poker Math
 

I was tinkering with this for a little while, but stopped when I realized how much time it would take (I need to get back to work).

One thing that did notice in your analysis:

[ QUOTE ]


Scenarios for a raise on the flop:

...

2) Villain calls down versus a raise on the flop and does improve on the turn/river

Hero contributes 3 SB preflop. Raise flop, 2 more SB. BB c/r turn (2 BB). Hero calls...



[/ QUOTE ]

We can't call this C/R on the turn because we know that BB has trips. I think we're drawing dead at this point because another Jack will just fill the villain up.

I could be wrong...

Actually, we do have the Jack outs if the turn is a 10, but only 2. I still don't think that's enough for a call.

Re: By Request, an Intro to Poker Math
 

flop/turn decisions are definitely math-intensive and very dependent on a possible future. Thanks for pointing out that error. I will try to correct it in a sec.

EDIT: Crap... too late to edit. I'll just post the correction in a new reply.

Re: By Request, an Intro to Poker Math
 

another quick note...

If Villain is soooted, there are several cards that can give him four to a flush on the turn. The number of these possible outs varies depending on which suit he is holding. (None if he has hearts)

The easiest way to account for this is to just say that our Villain considers finding a flush draw on the turn "unimproved" so he'll just fold anyway, but I don't find that likely.

I apologize if I'm picking nits. I only bring this up to see if I am over-complicating things. If I am, someone correct me so I can change my approach. I find these puzzles very interesting.

Re: By Request, an Intro to Poker Math
 

[ QUOTE ]
another quick note...

If Villain is soooted, there are several cards that can give him four to a flush on the turn. The number of these possible outs varies depending on which suit he is holding. (None if he has hearts)

The easiest way to account for this is to just say that our Villain considers finding a flush draw on the turn "unimproved" so he'll just fold anyway, but I don't find that likely.

I apologize if I'm picking nits. I only bring this up to see if I am over-complicating things. If I am, someone correct me so I can change my approach. I find these puzzles very interesting.

[/ QUOTE ]

that's one of those things that the person doing the calculations just doesn't really consider because it adds a lot of extra work. He will hope nobody else notices, or if they do that they won't point it out. This will add a lot of extra work so I probably wouldn't bother with it and just hope that it kind of evens out. This of course is incorrect, but much easier.

Re: By Request, an Intro to Poker Math
 

hahaha yea seriously... how much would my life suck if I had to take into account a BDFD?? Seriously I would have no time for poker if I were to throw that into the calculations. Anyway, something that comes in so rarely will have little impact on the final EV value.

Re: By Request, an Intro to Poker Math
 

Thank you for the clarification. I agree that trying to compute the implications of the backdoor flush is -EV.

Str8Fish, I hope I haven't come across as critical in any way. On the contrary, I have nothing but respect and appreciation for your work. I look forward to the remainder of your proof.

I'm just a long-time lurker trying to improve my understanding of the game by becoming more active on the boards. Please forgive my awkwardness.

Re: By Request, an Intro to Poker Math
 

The accuracy of these EV calculations SHOULD be analyzed as much as you are saying. Even small mistakes can lead to big errors in assumptions. I appreciate you putting the effort into understanding the math and being critical about it.

Re: By Request, an Intro to Poker Math
 

PART I- Raise the flop and go from there *Corrected for BDFD &amp; Scenario 2

Range of hands for villain:
88 – (6 combinations) [8[img]/images/graemlins/diamond.gif[/img]8[img]/images/graemlins/spade.gif[/img], 8[img]/images/graemlins/diamond.gif[/img]8[img]/images/graemlins/heart.gif[/img], 8[img]/images/graemlins/diamond.gif[/img]8[img]/images/graemlins/club.gif[/img], 8[img]/images/graemlins/spade.gif[/img]8[img]/images/graemlins/heart.gif[/img], 8[img]/images/graemlins/spade.gif[/img]8[img]/images/graemlins/club.gif[/img], 8[img]/images/graemlins/club.gif[/img]8[img]/images/graemlins/heart.gif[/img]] – calls down versus a raise
99 – (6 combos) - calls down versus a raise
TT – (3 combos, since hero has a T) – calls down versus a raise
JJ (1 combination – J[img]/images/graemlins/club.gif[/img]J[img]/images/graemlins/heart.gif[/img]) – villain 3-bets
QQ-AA – (18 combos) – 3-bets

ATs (3 combinations… cannot make 4 since hero has T) – villain calls, c/f the turn ui
AJs ( 2 combos with J on flop, J with hero) – 3-bets
AQs (4 combos) – calls and c/f the turn ui
AKs (4 combos) – calls and c/f the turn ui

AJo (4 Aces * 2 Jacks = 8 combos – 2 combos suited = 6 combos) – 3-bets
AQo (4 Aces * 4 Q’s = 16 combos – 4 combos suited = 12 combos) – calls and c/f the turn ui
AKo (16 combos – 4 combos suited = 12 combos) – calls and c/f the turn ui

KQs (4 combos) – calls and c/f the turn ui


Scenarios for a raise on the flop:

1) Villain calls down versus a raise on the flop and does not improve

Hero contributes 3SB preflop. Raising the flop he puts in 2 more. Then he puts in 4 more SB on the turn and river. = 4.5BB contributed to win a pot that is 6.25SB preflop, 10.25SB flop, 14.25SB turn, 16.25SB river = 8.125BB final total pot.

2) Villain calls down versus a raise on the flop and does improve on the turn/river and c/r’s hero.

Hero contributes 3 SB preflop. Raise flop, 2 more SB. BB c/r turn after improving (1 BB). Hero folds, hero loses 3.5BB.

Hero contributes 3 SB preflop. Raise flop, 2 more SB. Hero bets turn (1BB). BB c/r river after improving, hero calls (2BB). Hero loses 5.5B.


3) Villain calls raise on flop and c/f the turn ui.

Hero contributes 3 SB preflop. Raise flop, 2 more SB. BB c/f the turn ui. We contributed 5SB to win a 6.25SB preflop, 10.25SB flop = 5.125BB pot.

4) Villain 3-bets a raise on the flop and we fold the turn ui.

Hero contributes 3 SB preflop. Raise flop, get 3-bet, 3 more SB. BB bets turn and we fold. = -3BB.

5) Villain 3-bets a raise on the flop and we improve on the turn and call down.

Hero contributes 3 SB preflop. Raises flop, gets 3-bet, 3 more SB. BB bets turn, we call (1 BB), BB bets river, we call (1 BB). Hero contributes 5BB to win a pot that is 6.25SB preflop, 12.25SB flop, 16.25SB turn, 20.25SB river = 10.125BB pot.



EV calculations

Ok, so according to our hand range, we have:

15 combinations that plan to call down versus a raise.
27 combinations that 3-bet a raise on the flop.
36 combinations that intend to call and c/f the turn ui.
Sum = 75 combinations total

EV(raise) =
[villain calls down versus raise]
[88-TT] 15/78 * (8.125BB won * (43/45*42/44) [odds villain doesn’t improve with his 2 outer] – 3.5BB * (2/45)[villain improves on turn] – 5.5BB * (43/45)[villain doesn’t improve on turn] * (2/44)[villain improves on river]) +

[villain 3-bets a raise]
[JJ] 1/78 * (-3BB * (42/45)[no T on turn] – 5BB * (3/45)[T on turn]) +

[QQ-AA] 18/78 * (-3BB * (40/45)[no T/J on turn] + 10.125BB * (5/45)[T/J on turn] * (42/44)[no 2 outer on river] – 5BB * (5/45)[T/J on turn] * (2/44)[villain hits 2 outer]) +

[AJo] 6/78 * (-3BB * (42/45)[no T on turn] + 10.125BB * (3/45)[T on turn] * (40/44)[no J or A on river] – 5BB * (3/45)[T on turn] * (4/44)[J or A on river]) +

[AJs] 2/78 * (-3BB * (42/45)[no T on turn] + 10.125BB * (3/45)[T on turn] * (40/44)[no J or A on river] – 5BB * (3/45)[T on turn] * (4/44)[J or A on river] – 5BB * (1/45)[T[img]/images/graemlins/club.gif[/img] on turn] * (8/44)[8 remaining [img]/images/graemlins/club.gif[/img]’s on river] * (1/2)[1 possible BDF’s out of 2 suited AJ’s]) +

[villain c/f the turn ui, c/r if improved]
[AQo,AKo] 24/78 * (5.125BB * (39/45)[villain hits none of his 6 outs] – 3.5BB * (6/45)[villain hits one of his 6 outs on turn] – 5.5BB * (39/45)[villain misses turn] * (6/44)[villain hits river]) +

[AKs, AQs, KQs] 12/78 * (5.125BB * (~30/45)[villain hits none of his 6 outs or any of his suit – 9 for [img]/images/graemlins/club.gif[/img], [img]/images/graemlins/diamond.gif[/img], 10 for [img]/images/graemlins/spade.gif[/img]] * (3/4 possible flushes) + 5.125BB * (9/45)[villain hits suit on turn] * (30/44)[villain misses 6 outs and his suit on river] * (3/4 possible flushes) – 3.5BB * (6/45)[villain hits one of his 6 outs on turn, none his suit] – 5.5BB * (39/45)[villain misses 6 outs turn] * (6/44)[villain hits river] – 5.5BB * (9/45)[villain hits suit on turn] * (8/44)[villain hits suit on river] * (3/4 possible flushes)]

= [88-TT] +1.35BB + [JJ] -0.04BB + [QQ-AA] -0.37BB + [AJo] -0.17BB + [AJs] -0.06BB + [AQo,AKo] 1.02BB + [AQs, AKs, KQs] 0.28BB

EV(raising flop) = +2.01BB


Parts II and III are still on their way.