Deriving Pot Odds in No Limit Hold 'Em

Have you ever wondered where pot odds in Texas Hold 'Em comes from? Well let me show you.

Start with the formula for the expected value of being involved in a hand:

\[E[C] = p_{w}c_{w}-p_{l}c_{l} \]

where \(p_{w}\) is the probability of winning the pot, \(c_{w}\) is the pot size, \(p_{l}\) is the probability of losing the pot, and \(c_{l}\) is the immediate amount that you're risking/calling size. Notice that \(p_{l}\) is just \(1-p_{w}\) so if we make the substitution, the expression for \(E[C]\) becomes:

\[E[C] = p_{w}c_{w}-(1-p_{w})c_{l} \]

After distributing \(c_{l}\) to both terms,

\[E[C] = p_{w}c_{w}-c_{l} +p_{w}c_{l} \]

Combining like terms,

\[E[C] = p_{w}(c_{w}+c_{l})-c_{l}\]

Notice that for any situation we're faced with, we want to be in situations where the right hand side of the equation is greater than or equal to 0. In other words, we want to know under what conditions will we at the very least break even. Therefore, our equation then transforms into:

\[0 \le p_{w}(c_{w}+c_{l})-c_{l}\]

Rearranging our inequality yields:

\[p_{w} \ge \frac{c_{l}}{c_{w}+c_{l}}\]

The expression on the right-hand side is our beloved pot-odds fraction while the left-hand side is the probability that our hand will win the pot in front of us. To give you a concrete example, let's say you have KsQs (both spades) and the flop comes JsTs5d and you're facing heads up (just you and someone else). The pot started out with 10 BB and the villian puts in a bet of 5 BB. Pot odds gives you 33% but with 2 cards to still come, you have a flush draw and open-ended straight draw. Assuming that the villian just has top pair with no blockers, you have 15 outs which roughly translates to 45% equity. So even though you're behind at this point, the pot odds does not exceed your probability of winning the hand which means that calling is the profitable action.

And now you know where pot odds comes from! There are more advanced concepts that branch out from this but more on this later.