EV of a four-flush: intro

[DavidC]

You're in the button in a tight-aggressive game, with really weak blinds. Sklansky says you can raise any two cards with +EV here as a steal.

It's folded to you, you look down: 3s2s.

You raise. The SB folds and BB calls. Oh crap!

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Board is Ks9s8c, two of your suit. The defender leads into you. He has AKo, no backdoor flush.

(heads-up, pot is 5.5 SB).

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So you call, expecting that if you hit your hand you can raise and be called to the river.

If you hit on the turn, the pot will be 5.5 sb + 4 sb + 2 sb = 11.5 sb : 1sb for you to call.

Since you have a 4.22:1 shot at this, you're getting an EV of

+(11.5-4.22)/4.22 = +1.75 sb if you hit the turn.

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Supposing you miss the turn, but can raise if you hit the river and you will be called:

Evaluating the EV going from the turn to the river:

Pot will be 3bb, you're getting 5:1 odds on a 4.11:1 shot.

thats: +0.89/4.11 in EV (in bb this time): +0.21BB

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I'm going to post this part right now, just to hear a few yes/no's, and I'll continue by replying to this message.

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Edit: When replying to this message, please reply to the specific post instead of whatever post is at the bottom of the thread, as the subject lines will be changing slightly as new points are brought up.

[DavidC]

What I'm trying to do here is reconcile the implied odds on the flop and the implied odds on the turn to come up with a true effective odds answer here, to find the EV of the hand as played from the flop forwards.

If I were a better programmer with a working compiler, I would just run a couple hundred thousand hands through a simulator and see what the EV actually worked out to (within some small margin of error). Unfortunately, I'm not. [img]/images/graemlins/frown.gif[/img]

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One way of looking at the effective odds of this situation is:

1.5 BB in a 2.75BB pot (on the flop), with implied odds of 3 enemy bets if you hit:

5.75:1.5 on a 2:1 shot (in BB)

Now... I'm not sure 100% how to convert a x:1.5 into a x:1 ratio, but I THINK it's: (2x/3):1

If that's true, this gives us a ratio of 3.8333:1 on a 2:1 shot, giving us a EV of +1.8333/2 BB per hand, or +0.917 BB per hand.

[DavidC]

I'm a little uncomfortable with the effective odds answer given in the "effective odds intro", because it comes up with a number that's way higher than I would expect it, since we have such a small edge on the turn and a comparable edge to the effective odds intro answer, on the flop.

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So, if we did this, instead:

(chance of hitting turn) * (ev of flop to turn if we do so)...

(9/47) * (1.75/2)... {note: this is in BB now, rather than SB)

+ the chance of missing the turn * the ev of calling on the turn

+ (38/47) * 0.21

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=0.191 * 0.875 + 0.8085 * *0.21

=0.16755 + 0.169787

=~ 0.33734 bb/hand in +EV

This number is WAY different than the figure put forward by doing the 1.5 bb investment while getting about 2:1 odds.

Am I screwwing up the math somewhere here? How do we figure out how much money we're going to be making over 100k hands calling down to the river, raising and getting called if we do this play?

Keep in mind, I'm not looking at stuff like getting a free card, raising for folding equity when a guy has QQ or anything like that (maybe the chance of hitting your flush on the turn then on the river a fourth-flush card hits and he doesn't call very often on the river... that sort of thing too). I may get into that stuff later, but I'm trying to find out the answer to this (EV of this hand, starting at the flop and going to the river) first...

Any help would be appreciated.

--Dave.

Edit: FWIW, this number seems way too low.

[DavidC]

bump

[Raza]

I'm unsure about your BB per hand on this one. You are getting 3.8333 to 1 on a 2 to 1 shot so...

2 hands you will lose 1 and 1 hand you will gain 3.8333

2 * -1 + 1 * 3.8333 = 1.8333 BB per 3 hands.

So your EV per hand is .6111 BB. Yeah?

[Raza]

Ok, I worked this all out. Since I'm anal, I used the exact odds of 1.86 to 1 rather than 2 to 1.

To get that 1.86 to 1:

38/47 * 37/46 = .65
.65/(1-.65) = 1.86

Your effective odds are correct; you are getting 3.8333 to 1 on a 1.86 to 1 shot.

1 * 3.8333 + 1.86 * -1 = 1.97 BB per 2.86 hands
1.97 BB/ 2.86 hands = .69 BB per hand

Here's another way:

3 scenarios:
1) you hit on the turn
2) you hit on the river
3) you don't hit

1) you lose .5BB calling the flop and gain 5.75BB
the odds are 9/47
2) you lose 1.5BB calling the flop and turn and gain 5.75BB
the odds are 38/47 * 9/46
3) you lose 1.5BB calling the flop and turn and gain nothing
the odds are 38/47 * 37/46

EV = (9/47)(5.75BB - .5BB) + (38/47)(9/46)(5.75BB - 1.5BB) + (38/47)(37/46)(-1.5BB) = .702BB per hand

They are off by a bit. Maybe someone can point out where I erred, but I think this is about right.

[pzhon]

There are a few extra complications.

First, it is possible to win without hitting the flush. if the turn is a 2 or 3, you pick up an extra 5 outs.

Second, it is possible to lose with the flush, as the big blind can catch a runner-runner full house. If the turn is an offsuit ace, you only have 8 outs, and if the turn is a king, you have 7. If the turn is the ace of apades, the big blind has 4 outs.

The combination means you win 367/990 rather than the 360/990 that a flush would hit.

By the way, why would the big blind just call with AKo in the big blind? In an extremely tight game, I could imagine just calling against an early position raise, but not against a button raise.

[SocialWelfareIV]

The post confuses me a bit in terms of at what point it's calculating EV (is the flop bet/call in yet?). I'm going to calculate as if the flop bet has been made but the call has not.

DavidC's post specifies no backdoor flush draw, so we know 7 cards (3s, 2s, Ks, 9s, 8c, and an A and K that are not spades).

There are three outcomes: Hit on turn, hit on river, or miss. When we hit on the turn or river, we pick up 11.5 sb. When we whiff, we lose 1 sb on the flop and 2 sb on the turn, for a total of 3.

Probabilities of each outcome:
Hit On Turn: 9/45
Hit On River: 9/44
Miss: (36/45)(35/44)

If you want to calculate before flop call:
9/45 (11.5) + 9/44 (11.5) + (36/45)(35/44) (-3) =
2.272727 sb = 1.136364 BB