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01-07-2005, 02:14 PM
 Megenoita Senior Member Join Date: Oct 2004 Posts: 199
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:

http://www.twoplustwo.com/magazine/issue1/miller1.html

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.

M