Definition
Expected value is the long-term average outcome calculated by multiplying each possible result by its probability and adding them all together. When the future is uncertain, it gives you a rational benchmark for what you can expect to gain or lose on average over time.
How much could you make in a coin toss bet?
Imagine a simple coin toss game: flip heads and you win 1,000 won (about $0.75); flip tails and you get nothing. The probability of landing heads or tails is 50% (1/2) each. If you play this game just once, you either pocket 1,000 won or walk away empty-handed. You can never win 500 won on a single flip.
Yet, what happens if you play this game hundreds or thousands of times? Roughly half the flips will land on heads, earning you 1,000 won, and the other half will land on tails, giving you zero. Divide your total winnings by the total number of rounds, and you will find you earned an average of 500 won per game.
That 500 won is the expected value of the game. You calculate it simply by multiplying each outcome by its probability and adding them together. In short, it is an average weighted by the probability of each outcome.
Looking Closer: The Secret Behind Lotteries and Insurance
What if a lottery ticket costs 1,000 won, but its expected value is only 500 won? Buying a single ticket might land you a massive jackpot, but buying tickets repeatedly over time means you lose an average of 500 won per ticket. Most casino games are designed the exact same way—the player's expected value is lower than their bet, tipping the odds in favor of the house.
Insurance works in a similar, yet opposite way. Strictly by expected value, buying insurance is technically a losing financial bet. Premiums cover not just expected payouts but also company overhead and profit, meaning the expected payout is usually less than the premiums you pay. Even so, people willingly accept a negative expected return to buy peace of mind against catastrophic, ruinous risks.
To be precise, expected value is not a crystal ball predicting what happens next. Instead, it serves as a vital decision-making benchmark for evaluating whether a choice is advantageous or costly over the long run.
When Probability Meets the Law of Large Numbers
In a single trial, the outcome might never match the expected value itself. When you roll a standard six-sided die, the expected value is 3.5, yet you will never roll a 3.5. Expected value does not predict 'what comes up this time'—it tells you the 'average outcome across endless repetitions.'
What makes expected value so powerful is that as the number of trials grows, your actual average result reliably converges toward it. There may be lucky streaks and cold spells along the way, but the numbers inevitably cluster around this baseline. In mathematics and science, this principle is known as the Law of Large Numbers. It is the very reason casinos and insurance providers can run profitable, sustainable businesses year after year.
Keeping expected value in mind can sharpen your everyday choices. By weighing the potential gains and losses of an action against their probabilities, you can make clear-headed, rational decisions instead of acting on gut feelings. Just remember: if a potential loss is too catastrophic to bear, you must weigh the downside risk alongside the expected value.
Common misconceptions
Expected value is the exact number you expect to see on the next try.
Expected value is not the outcome of a single trial. Just as rolling a die has an expected value of 3.5 even though no side shows 3.5, it represents the average value that outcomes converge toward over countless repetitions.
Where you meet it
Lotteries pay out only a fraction of ticket sales as prizes, meaning that even with a multi-million-dollar jackpot, the expected value of a single ticket is almost always lower than its purchase price.
Because each face of a six-sided die has a 1 in 6 probability of landing face-up, the expected value of a single roll is (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5.
In one sentence
The average baseline outcome of an uncertain event over repeated trials, calculated by multiplying every possible payoff by its probability and adding them up.