I am studing
The mathematics of poker of Bill Chen, and I am at the chapter of pot odds and implied odds. I have this doubt: Everyone who have a little background on probability knows that the expected value is mathematical defined as
<x>=P1 x1 +P2 x2
P1+P2=1, that means
U is the whole universe of possible cases. When
xi are dollars we are calculcating the expected value of our possible gain. In the book of Chen I found many examples where, in the calculation of expected value of the possible dollars gain, the sum of the probability of the whole outcomes does not equate 1. From the mathematical point of view this is an error and can lead to a wrong answer. So my question is, which is quibble that I did not understand on the calculation of the expected value in a poker hand? Namely, is there some approximation that I did not understand? I guess that is an approx in order to perform quick calculations, but I am not sure.
Example: There are two players A and B. Player A hold A♠,A⋄, player B hold 9♥,8♥. On the board we have K♥,7♠,♥,2♥. The game is 30$-60$ texas-hold-em and the pot is 400$. Imagine that the player knows the cards that the opponents owns (is a conceptual exercise). 35/44 is the probability favorable to the player A and 9/44 is the probability favorable to the player B.
If the player A checks, we have (this is taken from the book)
<$(A)>=35/44 400$ ~318$
If the player A bet 60$
<$(A)>=35/44 (400$+60$+60$)- 60$ ~ 353$
We can see that in the second case the probability does not sum up to 1.