Cross Posted from Poker Theory:

I am trying to create an equation to show the expected value of a 3bet preflop followed by a cbet if called. The assumption is that our cards have no value and we always have zero equity and we are not allowed to barrel turns or rivers, so we are simply looking at bluff value of preflop and the flop.

The assumption is that villain opens the CO to 3bb, and we 3bet OTB to 9bb. The blinds always fold in this example. The villain folds 50% of the time, 4bets 10% of the time, and calls 40% of the time. When he calls, he will fold to our 10bb cbet 50% of the time. You may not agree with these numbers in the real world, but this is just for the sake of argument.

The equation I created was as follows:

EV = (% he folds)(amt won when he folds) + (% he calls)(amt won/lost when called) - (% he 4bets)(amt lost when he 4bets)

EV = (.50)(4.5bb) + (.40)(cbet expectation) - (.10)(9bb)

cbet expectation = (% he folds)(amt won when he folds) - (% he calls)(amt lost when he calls)
cbet expection = (.50)(19.5) - (.50)(10) = 4.75bb

EV of 3bet = (.50)(4.5bb) + (.40)(4.75bb) - (.10)(9bb)
EV of 3bet = 2.25bb + 1.9bb - 0.9bb
EV of 3bet = 3.25bb
aka 325bb/100 winrate in this spot

However, my friend did the math a different way, and is coming up with a much different result and I'm having a hard time refuting his way of calculating the results.

His method is to enumerate the various possibilities if we did this 100 times (assuming an even distribution) and list the profit/loss for each.

So, 50 times the villain folds pre. 10 times he 4bets pre. 20 times he calls pre and then folds on the flop. 20 times he calls pre and continues on the flop and we lose.

50 times our stack increases by 4.5bb = 225bb
10 times our stack decreases by 9bb = -90bb
20 times our stack increases by 10.5bb (he calls pre and folds flop) = 210bb
20 times our stack decreases by 19bb (our 3bet and failed cbet) = -380bb

This adds up to an expectation of -35bb given 100 trials, or a -0.35bb expectation for one occurrence.

Where does my math break down?

Help!

someone posted this in the other thread, and it seems to make sense to me

I get the same number as you -> EV=3,25

This is correct. I find it very helpful to think about EV at a decision point as the changes in your stacks after that specific decision point. So:

EV of 3bet = (.50)(4.5bb) + (.40)(4.75bb) + (.10)(-9bb) = -0.35bb

Here you can clearly see the mistake in the 2nd term : 1st and 3rd term are in measured in changes in stack BEFORE you put in 9bb, but 2nd terms is measured by changes in stack AFTER you put in 9bb (i.e post flop), thus the math mistake and the correct adjustment.
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Not nit picking or anything, I'm just a poker noob but figured to be decent at math. I came across this thread and hope to add some clarifications. Btw, as a long term winning reg, do you think this formula underestimate our EV in this situation?

I'm looking to open up my game a bit and 3betting BTN v.s CO open pretty wide seems to be a profitable spots. I would expect people to fold closer to 65%, 4b like 20% and rarely call 3b OOP so our hand wont matter that much. Leads me to the conclusion that 3betting with blockers is better.

thoughts?