I haven’t played poker for many years, but I thought I might get back into it, so I decided to write out a few theorems for myself that might be useful.

Basic Calling Theorem

Theorem: Suppose in a poker game that you have the option of folding or paying $B$ dollars to call a bet. Suppose that the pot contains $P$ dollars before you call. If you call, no more bets are permitted (e.g. either you or your opponent is “all-in” or you are heads up on the river), more cards are dealt, and the probability of you winning is $\alpha$. Then you should call if

$$\alpha > \frac{B}{B+P}.$$

Proof: The expected value if you call is

$$E = \alpha P + (1-\alpha) (-B)$$
since you will gain $P$ dollars if you win and lose $B$ dollars if you lose. Now the following are equivalent
$$
\begin{aligned}
E&>0 \\
\alpha P + (1-\alpha) (-B) &>0\\
\alpha P -B +\alpha B&>0 \\
\alpha (P+B) &>B \\
\alpha &>\frac{B}{P+B}. \\
\end{aligned}
$$
Thus, your expectation is greater than zero if and only if
$$ \alpha > \frac{B}{B+P}.$$

Example

Suppose that you are playing no limit Holdem poker, pre-flop a player goes all-in, and you are sure that he has a pair of aces, kings, queens, or jacks. You have seven six suited. All other players have folded. The pot contains \$130 and you need to pay \$90 to call. Should you call?

You have about a 21% chance to win. According to the Basic Calling Theorem, you should call if

$$\begin{aligned}
\alpha &> \frac{B}{B+P} \\
0.21 &>\frac{90}{90+130}\approx 0.4091.
\end{aligned}
$$
Your win probability $\alpha =0.21$ is not greater than $B/(B+P)\approx 0.4091$, so you should fold.