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Shai,
Thanks much for your reply.
Just so I'm sure I understand. We can bet turn and cb brick rivers and overall will make $90.20. Given the other assumptions. When we hit, it makes up for the times we brick. Basically. Right?
Not exactly. I'll try to explain. We do make money when we hit but we also make money from the turn bet. We have to look at the entire situation from the perspective of where we started: on the turn.
There are 3 turn events that make up the "sample space" of the solution, which means adding these events together gives us the solution. I'll break it down.
10% of the time we get CRAI OTT. We lose about $3.38 from this.
EV(CRAI) = (.1)*(.3116*455 - .6884*255) = -3.3764
15% of the time villain folds OTT, which is worth $27
EV(VFT) = .15*(180) = 27
The third event is much more complicated since it is multi-street. 75% of the time villain calls, which has various resulting events. We can think of the river events as their own sample space. Either hero checks behind, in which case he wins with J/8 or he loses with other rivers, or hero shoves the river, in which case villain either folds or calls and loses. We assume we never shove and lose, as this only happens if villain has a boat on exactly 2 river cards. We also assume villain doesn't lead the river. So the river sample space is broken into 2 events, hero checking or hero shoving.
From the perspective of being on the river, hero checking is worth -$42.53
EV(HChecksRiver|VcalledTurn) = 30/46*(6/46*300 - 40/46*120) = -42.5331
From the same perspective, shoving on the river is worth
EV(HShoveRiver|VcalledTurn) = 131.30
So the two river events sum to $88.77
But these take place in the river sample space. We have to weight the events properly so the three turn events (villain crai, villain folds, or villain calls) add to 1. And villain calls 75% OTT so they do add to 1 (.1 + .15 + .75 = 1)
So from the perspective of the turn, these river actions are worth only
EV(VCall) = .75*88.77 = $66.58. This is because we only get to the river 75% of the time.
So to add what the whole play is worth from the turn, we add the three turn events:
Total EV = EV(CRAI) + EV(VFold) + EV(VCall)
Total EV = -3.38 + 27 + 66.58 = $90.20
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Oh and btw this doesn't even account for the times our j high is good. Which we do see once in a while. Obv immaterial amount of times, but does happen.
Thanks again.
No problem. I did partly but not completely accurately account for the Js and 8s. I assumed we'd win with a J or 8 when it got checked back, or 1 in 6 times we shove with these. Basically I estimated they were worth 1 out. They're probably worth a bit more if villain is stationy. If we assume he's super stationy, to the point our Js and 8s are always good, then the result is worth $103.96, (note if we're shoving these hands we can't win them when checking them back so I have to eliminate those terms).
This isn't entirely accurate though as he's not always going to call. I can refine the model more but adding terms mostly just complicates things. Do we care about the actual EV or which play is better? Unless the EV of separate plays is very close, we generally shouldn't worry about accounting for variables that only slightly change the result.
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Shai,
What is the EV of this same scenario, but just change to: when we hit and ship, he calls zero % of the time (rather than 50%).
(I assume this won't take but a second for you to plug in that scenario. If you don't have your calculation saved and have to redo, then you don't have to do it all again. NM).
Just curious. Trying to get a better "visual" of what's going on in this type scenario.
Well it's not good for us if he never calls when we hit, but we still make a lot of money from him folding. I'm going to assume we're still sometimes good when it gets checked back so I don't have to change the model, but if we drop his calling rate to 0 when we shove the total EV is $69.98. We still make money from villain folding so much, but now the line is as bad as checking the turn and bet/checking depending whether we hit.
However this is not realistic. Compare a river fold rate of 80% where we make $78.07.
Also, if villain is folding the river so much, he will fold more on the turn also. Let's say he folds on the river when we hit 80% and folds on the turn 30% (twice as often as he was). Now the line is worth $94.18. So the play is fine if he folds a lot in general, but if he's always calling the turn only to fold when a heart or 3-straight hits, it's not so good.
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Originally Posted by winky51
I think why it is so hard to understand is the short hand format he is using. I get it. I do work like this all the time for scenarios.
P(X) or EV(Y) is not shorthand exactly. More notation. They mean "Probability of Event X" and "Expected value of random variable Y" respectively. One could consider it shorthand except it's the standard way to write out functions. Not doing it like this is super clunky. Which I think you get, but just pointing this out.
I think these EVs are hard to calculate mainly because it's difficult to account for 30+ variables accurately and then input them correctly into a computational engine. Most people make numerous mistakes even if they understand the fundamentals. When I taught calculus I saw this a LOT--students who I knew understood the concepts and even set up the integrals correctly with the correct steps, but how many do you think ended up with the correct result after a page-long problem?
Maybe 1 in 100. Seriously. I didn't even count off for the results being wrong if the solution was correct because otherwise 198/200 of my students would have failed.
I make mistakes like this occasionally but far less (multiple orders of magnitude) than most people for some reason. I'm not necessarily smarter--I just work carefully and methodically and assume I messed up until I've checked two or three times that I haven't. As a kid I'd usually be one of the first to finish my exams, but always the last to turn them in because I'd just check everything over for as much time as I had.