What is expected value?
Expected value is the probability multiplied by the value of each outcome. For example, a 50% chance of winning $100 is worth $50 to you (if you don’t mind the risk). We can use this framework to work out if you should play the lottery. Let’s say a ticket costs $10, and you have a 0.0000001 chance of winning $10 million dollars — should you buy one? Without using expected value, this is a nearly impossible question to evaluate. The value to you of having one of these tickets is $1 (0.0000001 x 10,000,000) but costs you $10, so it has negative expected value. This is true of most lotteries in real life, buying a lottery ticket is just an example of our bias towards excessive optimism.
Examples of using expected value
It turns out that all events have some aspect of risk and value. Insurance companies use this to determine how much to charge you for your premiums. They add up everyone in your reference class, and determine how much it costs them on average in payouts. They then add a little on the top to make a profit, which makes buying insurance net negative (the costs minus the benefits to you) on expectation, just like buying a lottery ticket. However, this doesn’t mean getting insurance is a bad idea! A lot of people don’t like taking on excessive risk (a small chance of becoming bankrupt feels much worse than paying up for insurance you might never need), so buying insurance is rational. Another way to put this is that we have diminishing marginal returns to extra money (or concave utility functions, for the mathematically inclined).
Pascal’s wager is also an example of using expected value to think about the world. Humans all bet with their lives either that God exists or that he does not. Pascal argues that a rational person should live as though God exists and seek to believe in God. If God does actually exist, such a person will have only a finite loss (some pleasures, luxury, etc.), whereas they stand to receive infinite gains (as represented by eternity in Heaven) and avoid infinite losses (eternity in Hell).