The Fundamental Theorem of Implied Odds

[Bad Lobster]

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Furthermore, there is some chance Q that when you win, you will win a pot of 2 S, representing a gain of S. Therefore your expectation is at least:

P x Q x S

Now we want positive expectation, which means the reward of calling must exceed its cost, so we want:

B < P Q S which will hold iff

S > B / (P Q)



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There's a hidden assumption here.

Even if we assume that Q (the chance opponent will call your huge bet) is always greater than 0, your formula requires that either Q is fixed or there is a minimum positive value of Q. But if Q is a function of your bet size B, it's possible for Q to always be positive and still have no minimum.

For example, suppose the probability your opponent will call a bet of B dollars without having the nuts himself is 1/B. There's a 1/1000 chance he'll call a $1000 bet, a 1/1 million chance he'll call a million dollar bet, and so on. Then you can see that no matter how big your bet, your expected profit stays around 1 dollar.

This sort of situation happens in a lot of mathematical proofs: there's a difference between "There is an X such that for every Y..." and "For every Y, there is an X such that..." It seems like an obscure distinction but it can often make the difference between a proposition being right or wrong.

This is more in line with common sense, too--you can't really expect the chance your opponent will call a 1 million dollar bet to be a serious possibility.