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Old 01-07-2005, 02:14 PM
Megenoita Megenoita is offline
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Default Ed Miller\'s Response to Briar Probability Error???

Really, I'm asking where my error is-I'm pretty new to this so please be merciful. For this question, I am referring to "Responding to Jim Briar" by Ed Miller along with page 109 of SSHE. The online article is here:

In the K [img]/images/graemlins/diamond.gif[/img] J [img]/images/graemlins/diamond.gif[/img] hand analysis, Miller writes:

Call the probability that your opponent is bluffing or betting a weaker hand, x, that he has two pair or a better one pair hand, y, and that he has a set, z. Also, assume that you have an 80 percent chance to win if he is bluffing, a 20 percent chance if he has two pair or a better one pair hand, and no chance against a set. Your expectation for calling down is thus:

EV = (0.80x)(13) + (0.20y)(13) + (0.20x)(-2) + (0.80y)(-2) + (z)(-2)

In the book, I enumerate the possible two pair and set hands your opponent can likely have. Using that enumeration, he can have two pair any of 27 ways and a set any of 7 ways. Thus, the ratio of the probabilities that he has two pair to a set, y/z, should be approximately 27/7. Using this ratio, it is correct to call down automatically even if your opponent never bluffs. That is, setting x = 0, y = 27/34, and z = 7/34:

EV = 0 + (0.159)(13) + 0 + (0.635)(-2) + (0.210)(-2) = 2.07 - 1.69 = 0.38, greater than zero.

I've spent 3 hours trying to understand this, but in the first paragraph above Miller states that you have 20% equity if your opponent has either 2 pair OR A BETTER ONE PAIR HAND, however, when he writes his equation, he doesn't seem to account for the 1 pair probabilities--he says that the ratio of 2 pair to a set is 27/7, or about 80%, and substitutes that figure for y, but this does not consider the 1 pair-better kicker possibilities (which would make y = 87.5%). Therefore, he is considering the equity for both 2 pair and better 1 pair hands, but not the probability. It seems that he either has to lower the equity to be limited to ONLY when you are against 2 pair or a set, OR include the 1 pair-better kicker probability in the equation. I calculated that your equity is only 14.5% against the 2 pair possibilities:

K9 (6): 3
K5 (6): 6
95 (9): 8
K3 (6): 9

Total 180/27 = 6.67 outs or 14.5%

So my equation for EV against 2 pair or a set if you call down (with no chance of him bluffing or having a worse hand):

EV = 0 + (.145)(.79)(13) + 0 + (.86)(.79)(-2) + (.21)(-2) = -.28

My equation if you include the better top pair figures:

EV = 0 + (.22)(.88)(13) + 0 + (.78)(.88)(-2) + (.125)(-2) = .89

Further, regarding when Ed says:

Using this ratio, it is correct to call down automatically even if your opponent never bluffs.

I don't see why you would ever call the river bet when you don't improve if you are saying that he NEVER bluffs...this means you KNOW that you have lost on the river since all that he can have is 2 pair or a set, or top pair with a better kicker (we have not given him the possibility of bluffing or having a worse hand). It may be +EV to call down, but it is not correct.

Could someone who understands this kind of thing please explain to me where I err? All instructive comments are much appreciated.

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Old 01-07-2005, 04:45 PM
Ed Miller Ed Miller is offline
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Default Re: Ed Miller\'s Response to Briar Probability Error???

You aren't wrong on either count. Most of this confusion is due to a little sloppiness on my part.

On your first point, that I include better one pair hands in one place but not another... you should understand that I am making a lot of approximations in this calculation. I'm making one huge approximation regarding the range of hands your opponent will have (the 27/7 approximation, and then again with the 1/1 approximation).

In other words, I know my opponent's range of hands so inexactly that it does me no good to calculate the rest of the parameters exactly. There are two main things I want people to glean from this calculation:

1. This is the general mathematical approach you should use to get the answer. You can make approximations if you like, or you can try to be as exact as possible. Just remember the process and that the accuracy of your results will vary depending on the assumptions you make.

2. Calling that turn raise becomes correct at a surprisingly low bluffing percentage. While I didn't try to crunch exact numbers, I don't have to to prove this point. If you change a few of the assumptions, my 5% bluffing rate might drop to 3% or rise to 8%. But it isn't going to rise to 30% unless you make some very strange assumptions. And that's really the point of my article. Don't say, "He can't be bluffing" when you really mean, "Over 90% of the time he won't be bluffing."

So in other words, the article is merely designed to spark your interest, not prove a result beyond a shadow of a doubt. You taking the math and running with it is great. Just make sure you understand the large role your approximations and assumptions play in getting results...

On your second point, that it isn't right to call a river bet if you know 100% you are behind... well, obviously you are right. That was sloppiness of wording on my part... something I should have caught. What I intended to say was, "Calling down automatically is better than folding to the turn raise." It's better to call and then call again on the river (throwing away a bet with zero chance to win) than it is to fold to the turn raise. That's because your chance to draw out against two pair on the river card is worth more than the bet you sacrifice on the river.

But it may be better still GIVEN SOME STRANGE ASSUMPTIONS (like 0% bluffing frequency) to call on the turn and fold on the river.

However, IN PRACTICE, that is rarely correct. Usually against typical opponents you should call and call again.

I hope that clears up your confusion.
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Old 01-07-2005, 05:56 PM
Megenoita Megenoita is offline
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Join Date: Oct 2004
Posts: 199
Default Re: Ed Miller\'s Response to Briar Probability Error???

Thanks, it really helps clear things up. I understand your point in the article, and it is well taken; my confusion was due to not knowing how precisely you were taking the math, as I'm trying to learn these "simple" calculations myself.

The equation is really helpful in my game as the greatest leak I'm trying to plug right now is both betting on the river (knowing if I will call a raise) and calling down when there is a turn the article is pertinent to my growth at the moment.

Thanks again,
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