Mathematics
Expected Value
Quick fact
In a fair coin flip, the expected value of winning $1 for heads and losing $1 for tails is $0—the game is 'fair' because the long-run average is exactly zero.
Why this is interesting
Imagine you’re offered a coin flip: heads you win $100, tails you lose $50. Would you play? Intuition might say yes, but the real answer lies in a simple calculation that reveals the 'expected' payoff—and it might surprise you.
Read the full explanation
Understanding Expected Value
Expected value is the average outcome you’d get if you repeated an experiment many, many times. For a coin flip with $100 for heads and $0 for tails, half the time you get $100, half the time $0. The expected value is ($100 × 0.5) + ($0 × 0.5) = $50. That doesn’t mean you’ll ever get $50—it’s the average over hundreds of flips. Think of it like this: if you played that coin flip a thousand times, you’d expect to end up with about $50,000 total, or $50 per flip. Expected value weights each possible outcome by its probability, giving a single number that represents the 'center' of the distribution.
A deeper explanation
Formally, expected value is the probability-weighted sum of all possible outcomes: E[X] = Σ xi P(xi). This works for discrete and continuous outcomes (using integrals). The power of expected value lies in its linearity: E[aX + bY] = aE[X] + bE[Y], even if the random variables are dependent. This makes it incredibly useful in finance, insurance, and game theory. For example, an insurance company collects premiums that exceed the expected payout, ensuring profit on average. However, expected value alone doesn’t capture risk—a game with a 50% chance to win $1,000,000 and 50% to lose $900,000 has a positive expected value of $50,000, but the huge potential loss might be unacceptable. That’s why concepts like variance and expected utility refine the picture. Expected value is the foundation for all these ideas, serving as the rational anchor in a world of randomness.