So far, we've talked about expectation as the average outcome you'd get from something like rolling a die many times. That's useful, but the real power of expectation is in decision-making. Life is full of uncertainty. Should you take a new job? Invest in a stock? Pack an umbrella? Expectation gives us a mathematical framework for making the best possible choice when we don't know for sure what will happen.

The key is to think not just about the value of an outcome, but its utility. Utility is a measure of the satisfaction or usefulness you get from an outcome. A gain of $100 might mean a lot to a student, but very little to a billionaire. They have different utilities for the same amount of money. By calculating the expected utility of each choice, we can systematically pick the one that offers the most satisfaction on average.

This lets us compare apples and oranges. We can weigh a safe bet with a modest reward against a risky bet with a huge potential payoff. The math tells us which path is logically better in the long run.

Let's make this concrete. Imagine you're offered a part-time job. You have two payment options:

1. Guaranteed Pay: You receive $150 for your work.
2. Commission: You have a 50% chance of making $400 and a 50% chance of making $0, depending on sales.

Which option should you choose? Let's assume for this simple case that utility is equal to the dollar amount.

For Option 1, the outcome is certain. The probability is 100% (or 1), and the utility is 150. So, the expected utility is: $E[U_{Option1}] = 1 \times 150 = 150$

For Option 2, we use the formula: $E[U_{Option2}] = (0.5 \times 400) + (0.5 \times 0) = 200 + 0 = 200$

Based on expected utility, the commission-based plan is the better choice. Even though it's risky, its average payoff is higher.

Let's look at a more complex problem. Imagine you are presented with a series of potential rewards, one by one. You know the probability distribution of the rewards, but you don't know the order they will appear in.

Here are the rules:

• You see one reward at a time.
• For each reward, you can either accept it and the game ends, or reject it and see the next one.
• You cannot go back to a reward you've rejected.

This is called the Prophet Inequality problem. The name comes from comparing the outcome of your strategy to that of a 'prophet' who can see all the rewards in advance and simply pick the best one. The inequality shows that a simple strategy can get you surprisingly close to the prophet's perfect outcome.

A remarkably effective strategy is to set a threshold. You calculate the expected value of the next prize, and you accept the current prize only if it's higher than that expectation. Let's say you have two prizes left, drawn from a distribution with an average value of $50.

When you see the first of these two prizes, you should only accept it if its value is greater than $50. Why? Because by rejecting it, you can expect to get $50 on average from the last prize. If the current prize is better than that expectation, you take it. Otherwise, you take your chances on the next one. This type of reasoning, balancing current information against future expectation, is at the heart of decision theory.

Using expectation as a guide helps us navigate uncertainty and make choices that, on average, will lead to better outcomes. It's a powerful tool for thinking logically about the future.