Expected Value
The long-run average score you can expect to walk away with if you play an uncertain game over and over again.
Definition Expected value is the weighted average of all possible outcomes of an event, calculated by multiplying each outcome by its probability of occurring and adding them together. When outcomes are uncertain, it provides a rational baseline for what you can expect on average over the long run.
How Much Can You Expect from a Coin Toss?
Imagine a coin toss game where you win $1 if it lands on heads and $0 if it lands on tails. The chance of getting heads or tails is equal: 1 in 2 (50%). If you play this game just once, you either walk away with $1 or nothing at all. You can never actually receive 50 cents in a single flip.
What happens, though, if you play this game 100 or 1,000 times? About half the time you will get heads and win $1, and the other half you will get tails and win $0. If you divide your total winnings by the number of rounds, you will end up with an average of 50 cents per game.
That 50 cents is the expected value of the game. You calculate it by multiplying each possible outcome by its probability and adding them up. In short, it is an average weighted by the probability of each outcome occurring.
A Closer Look: The Secret Behind Lotteries and Insurance
What if a lottery ticket costs $2, but its expected value is only $1? Even though a lucky winner might take home millions in a single drawing, buying tickets repeatedly over time means you are losing an average of $1 per ticket. In fact, every game in a casino is designed so that the player's expected value is always lower than the cost to play.
Insurance, on the other hand, might look like a losing bet strictly in terms of expected value. Because major disasters are rare, the expected payout is typically lower than the total premiums you pay. However, people willingly choose stability over expected value loss to protect themselves against catastrophic risks they could never afford alone.
To be precise, expected value is not just a tool for predicting a single outcome. It acts as the most objective compass for judging whether a decision is favorable or unfavorable over the long run in the face of uncertainty.
When Expected Value Meets the Law of Large Numbers
In a single trial, you might never actually see the exact number calculated as the expected value. For instance, the expected value of rolling a standard six-sided die is 3.5, but there is no 3.5 on any face of the die. Expected value does not mean 'the exact outcome you will get next time'; it means 'the average outcome after countless repetitions.'
Even so, expected value is an incredibly powerful tool because the real-world average gets closer and closer to it as the number of attempts increases. In science and mathematics, this is known as the Law of Large Numbers. It is the very principle that keeps casinos and insurance companies profitably in business.
You can apply expected value to everyday decisions too. Weigh the potential gain against the potential loss for any action, multiplying each by its probability. By using expected value rather than relying on gut feelings, you can make far more rational and grounded choices.
π€ Common misconceptions
The expected value is the specific number you should expect to see on your next attempt.
Expected value is not the outcome of a single trial. Just as you can never roll a 3.5 on a regular die, expected value represents the average outcome that results after countless repetitions.
π§Ί Where you meet it
The sum of all possible outcomes multiplied by their respective probabilities, serving as a rational baseline for the long-term average.