MOP Pg. 177
Posted by AF3
Posted by
AF3
posted in
Mid Stakes
MOP Pg. 177
Is it just me or is there an algebraic mistake at the top of page 177?
The authors claim:
(1/(1+q)) -1 + k = (k - q + kq)/ ((k)(1+q))
However, multiplying the LHS by (1+q)/(1+q) where applicable gives:
(1/(1+q)) -1 + k = (k + kq - q) / (1+q)
Why are they including an extra factor of k in the denominator? I mean, if you use real numbers for the parameters then the second formula gives the correct equality. It's kind of important because they go on to use the first formula in deriving the number of total kings that we check, but it seems like this formula is wrong.
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Can you elaborate on what you did here? edit: I see now, real reply below
Can you elaborate on what you did here?
First, I gave the claim that the authors make regarding an equality that is found at the top of pg. 177 in Mathematics of Poker.
Then, I claimed that this inequality was false by showing that the left hand side of that equation, when calculated properly, yields a value than they claim it does.
Right ok I see now sorry. So you're right (1/(1+q)) - 1 + k = (k+kq-q)/(1+q), but you call that quantity of the k kings so you divide the expression by k
The authors claim that both sides of the equation should be equal. AF3 multiplied the left hand side of the equation with (1-q)(1-q) to simplify the term:
(1/(1+q)) -1 + k
=> (1/(1+q)) - (1+q)/(1+q) + (k(1+q))/(1+q)
=> (1/(1+q)) - (1+q)/(1+q) + (k + kq)/(1+q)
=> (1-1-q+k+kq)/(1+q)
=> (-q+k+kq)/(1+q)
... which is clearly not the same as (k - q + kq)/ ((k)(1+q)), like the authors claim.
I have to admit that I´m confused as well ... I can´t see where we made a mistake?! Have no time atm, but will have a deeper look tomorrow.
read my post right above yours :)
Yeah, thanks. Found out the same and just wanted to post - but you were way faster. :D
Right ok I see now sorry. So you're right (1/(1+q)) - 1 + k = (k+kq-q)/(1+q), but you call that quantity of the k kings so you divide the expression by k
Thanks Steve, I got it. It just comes from the fact that we are normalizing our calling frequency by the amount of kings we have in the checking range, which is where we get the factor of 1/k.
Do you understand all of Pg. 177? The authors basically do some stuff in the last few paragraphs that they don't really explain and then literally say "With that...we have the optimal solution to the game".
They cover the strategy for Aces and Kings, but I don't (even) see where they get numbers for that from. When I use the ratios at the top of the page and then apply them to the numbers they derive for total number of hands bet and total number of hands and checked, I don't even get the same percentages.
Did they just mess up with the subscripts on some of the results?
For example:
The fraction of checking hands which are kings ~ .57957
The number of total hands bet is 1.45595
However:
Percentage of (All) Kings That Player X Checks = 84.16% = .57957 * 1.45595
= Fraction of checking hands which are kings * Total Number of Hands Bet
!!?
They just made a typo with what they called the number of hands bet vs. the number of hands checked, right?
There are actually two typos in that end part. The first is minor, they have 0.55405 when it should be 0.54405 (b+c has to equal 2). More importantly, they have b and c reversed. If you solve their system of equations you get b=0.54405 and c=1.45595. This makes sense as Ak+Kk=1.45595 and Ab+Kb=0.54405.
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