This is a distillation of this post by John Wentworth.

Introduction

Suppose you're playing a poker game. You're an excellent poker player (though you've never studied probability), and your goal is to maximize your winnings.

Your opponent is about to raise, call, or fold, and you start thinking ahead.

If your opponent raises, he either has a strong hand or is bluffing. In this situation, your poker intuition tells you he would be bluffing and you should call in response.
If your opponent calls, he probably has a better hand than yours.
If your opponent folds, you win the hand without need for further action.

Let's break down your thinking in the case where your opponent raises. Your thought process is something like this:

If he raises, you want to take the action that maximizes your expected winnings.
You want to make the decision that's best in the worlds where he would raise. You don't care about the worlds where he wouldn't raise, because we're currently making the assumption that he raises.
Your poker intuition tells you that the worlds where he would raise are mostly the ones where he is bluffing. In these worlds your winnings are maximized by calling. So you decide the optimal policy if he raises is to call.

Step 2 is the important one here. Let's unpack it further.

You don't know your opponent's actual hand or what he will do. But you're currently thinking about what to do if he raises.
The optimal decision here depends only on worlds where he would raise.
You decide how much you care about winning in different worlds precisely by thinking "how likely is this world, given that he raises?".

This sounds suspiciously like you're maximizing the Bayesian conditional expectation of your winnings: the expected value given some partial information about the world. This can be precisely defined as $E [u (A, X) | opponent raises] = \sum_{X s.t. opponent raises} P [X] u (A, X)$ , where $u$ is your winnings, $A$ is your action, and $P [X]$ is the probability of world $X$ . But you don't know any probability, so you don't know how to assign probability to worlds, much less what conditioning and expectation are! How could you possibly be maximizing a "conditional expectation"?

Luckily, your opponent folds and you win the hand. You resolve to (a) study coherence theorems and probability so you know the Law behind optimal poker strategy, and (b) figure out wh...

Pareto Efficiency