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I am reading a book called The Mathematics of Poker (Chen, Bill, and Jerrod Ankenman. The Mathematics of Poker. 1st ed. Pittsburgh, Pa: ConJelCo LLC, 2006. ).
I am a bit confused about Pot ODD and Probability. On page 51 Example 4.3:
B should call if his chance of winning the pot is greater than 30/105. Is chance here mean probability of winning? Or does it mean the ODD of winning?
I'm not sure whether you've described this example correctly, but the maths isn't correct.
If AK bets $30 into a pot of $75, there would be $105 in the pot, and if B calls this would rise to $135. In other words, B is risking $30 to try to win a total of $135, so he needs to win (on average) 30/135 of the time (not 30/105).
This is effectively a probability. If B will win the hand (on average) at least this often, he should call. You don't say which street this hand is from, but if (say) it's a turn decision then B's 8 outs give him only an 8/46 chance of winning the pot. This is less than 30/135 so he should fold (assuming he is basing the call purely on the probability of "hitting" on the river).
The term "pot odds" is more commonly expressed as a ratio, x:y (pronounced "x to y"). To take a simple example: suppose the pot is $1 and A bets $1. Now B's pot odds are 2:1 (x = the pot size before calling; y = the required call size). This means that B needs to win the pot at least once for every two times he loses it - in other words, he must win at least one time in three, or 1/3. This is the equivalent probability for pot odds of 2:1.
To apply this second approach to your example, the pot odds are 105:30. To calculate the equivalent probability, divide the call size by the sum of the "odds" ratio numbers, i.e. 30/(105+30) = 30/135.
The difference between pot odds and probability is essentially the same as the difference between fractional odds and decimal odds in bookmaking. Fractional odds of "2 to 1" are equivalent to decimal odds of 3 (normally written 3.0). But fractional odds are normally written as e.g. 2/1 ("2 to 1"), which might partly explain the confusion here: the equivalent probability is 1/3, not 1/2.
