By Request, an Intro to Poker Math

[Str8Fish]

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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.

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

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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.

[Str8Fish]

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.

[Finch]

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:

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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...



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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.

[Str8Fish]

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.

[Finch]

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.

[jaxUp]

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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.

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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.

[Str8Fish]

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.

[Finch]

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.

[Str8Fish]

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.

[Str8Fish]

PART I- Raise the flop and go from there *Corrected for BDFD & 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.