Two Plus Two Older Archives - Flop call with nothing but backdoors math.

After discussing the problem of calling a flop with nothing but 2 backdoor draws here http://forumserver.twoplustwo.com/sh...14&fpart=1 i decide to count more precisely if i like this thing. Guess you'll find results interesting a bit.

I repost a hand which i analyzed here as well as results:

Party Poker 30/60 Hold'em (8 handed) converter

Preflop: PeterRus is BB with 6[img]/images/graemlins/spade.gif[/img], 4[img]/images/graemlins/spade.gif[/img].
UTG raises, UTG+1 calls, 2 folds, CO calls, 1 fold, SB calls, PeterRus calls.

Flop: (10 SB) 5[img]/images/graemlins/diamond.gif[/img], T[img]/images/graemlins/spade.gif[/img], Q[img]/images/graemlins/heart.gif[/img] (5 players)
SB checks, PeterRus checks, UTG bets, UTG+1 calls, CO calls, SB calls, PeterRus calls.

Turn: (7.50 BB) 7[img]/images/graemlins/heart.gif[/img] (5 players)
SB checks, PeterRus checks, UTG checks, UTG+1 bets, CO calls, SB folds, PeterRus calls, UTG calls.

River: (11.50 BB) 3[img]/images/graemlins/diamond.gif[/img] (4 players)
PeterRus checks, UTG checks, UTG+1 checks, CO bets, PeterRus calls, UTG folds, UTG+1 calls.

Final Pot: 14.50 BB

Results in white below: 
PeterRus has 6s 4s (straight, seven high).
UTG+1 has Js Tc (one pair, tens).
CO has Kh Qs (one pair, queens).
Outcome: PeterRus wins 14.50 BB. 

The math on flop call:
I can catch outs on turn:

Strong draws.

1. 6 outs to simple 4-flush.
Probability to complete on river=9/46
Probability from flop to river=6/47*9/46=0.025

2. 2 outs to 4-flush with a gut shot.
Probability to complete on river=12/46
Probability from flop to river=2/47*12/46=0.011

3. 2 outs to 4-flush with open-ended 4-straight.
Probability to complete on river=15/46
Probability from flop to river=2/47*15/46=0.014

4. 6 outs to open-ended 4-straight.
Probability to complete on river=8/46
Probability from flop to river=6/47*8/46=0.022

Weak draws:

5. 6 outs to a gut shot.
Probability to complete on river = 4/46
Probability from flop to river=6/47*4/46=0.011

6. 6 outs to a simple pair.
Probability to complete to 2pairs/trips at river=5/46
Probability from flop to river=6/47*5/46=0.014

'I' - means implyed odds.

Scenario 1. OMG, free card on turn.

I mustn't pay additional bet on turn. It's also obvious, that i can count additional draws of a gutshot and 2pairs/trips as they will be best most of the time if completed and turn was checked around. So in this case my EV of flop call is:

(P1+P2+P3+P4+P5+P6)*(14+I)-(1-P1-P2-P3-P4-P5-P6)*1= 0.097*(14+I)-0.903. Without implyed odds equity will be 1.36-0.9=+0.46SB/h.

My estimate on implyed odds here = 1.2bb, so EV will be +0.68SB

Scenario 2. I must pay single bet on turn.

I'm planning to muck a gutshot and 2pairs draws for a single bet on turn except cases when im again closing the action and there is 1 bettor + at least 1 caller are ahead of me. So let's don't count a gutshot here cause it will be not often.

Probability to catch a strong draw is 16/47=0.34. In the rest of 0.66 cases i will loose only my flop bet.

So in this case my EV of flop call is:

(P1+P2+P3+P4)*(16+I)-0.66*1-(0.34-P1-P2-P3-P4)*3=0.072*(16+I)-1.464

Without implyed odds equity will be 1.152-1.464=-0.31sb/h

We can easily count how much bets we must collect to make this second scenario positive (we disclude single bet on turn from anyone as without this bet we come to scenario first). Borderline implyed odds equals 2 bigbets. The more bets to collect - the more ev. In my hand i collected 4 big bets, so my EV was 0.26SB/h.

My estimate on implyed odds in this scenario with 5 people to turn is actually more than these 4 big bets. It's around 5bb's i guess. So EV should be looks like +0.41SB/h

Scenario 3. Someone slowplayed and c/r-ed turn or someone tries to defend his hand on turn. All folds except us (3bb's of opponents money involved in a pot) We now must pay 2 big bets for our draw.

(P1+P2+P3+P4)*(20+I)-0.66*1-(0.34-P1-P2-P3-P4)*5=0.072*(22+I)-2.001

Without implyed odds equity will be 1.44-2.001=-0.56sb/h

In case we will pay for our draw 2 big bets on turn - we need to collect additionally to money for initial bet and raise - 4 bigbets to make our flop call borderline.

So if someone c/rses and initial bettor calls - we need to get 3 bb's on river to have our share. Not that impossible, right?

I guess usually implied odds here will be around 4 bb’s, so actual EV will be zero.

Scenario 4. Turn war.
2 people have excellent hands simultaneously. Turn is capped 3-way including us.

(P1+P2+P3+P4)*(30+I)-0.66*1-(0.34-P1-P2-P3-P4)*9=0.072*(30+I)-3.075

Without implyed odds equity will be 2.16-3.075=-0.92SB/h.

And we need to collect 6 big bets additionally at river which looks like impossible. I guess in average we will get 4.5BB and will have EV=-0.27SB/h

Ok, here we are. I discounted gut shot on turn, though it's obviously occasionally becomes positive, but i also discounted time we will be beaten by higher flush/full house/flush over our straight.

Now we can choose probability for scenarios and find EV of flop call. In my experience probabilities at 30/60 limit will looks like:

1 - 5%
2 - 40%
3 - 45%
4 - 10%

So final cypher of EV of our call will be: 0.05*0.68+0.4*0.41+0.45*0-0.1*0.27=0.17SB/h or 0.085bb/h

Which is very marginal. So even if there is a thin chance on flop that you'll be raised behind and pay on flop more than one bet - it's easy muck, regardless of implyed odds as they're dead already. Ok, if pot is already 20sb's - you can proceed. [img]/images/graemlins/smile.gif[/img]