Expected Value

In the last lesson the mean was a simple average of numbers you already had. Expected value takes that idea and applies it to an uncertain future, where each outcome has a probability instead of just appearing once. It answers the question: on average, what should I expect to happen? It is one of the more practical ideas in probability, because it is how gamblers, insurers, and investors decide whether a risky bet is worth taking.

The definition

The expected value of a random variable is the probability-weighted average of all its possible outcomes. You compute it by multiplying each outcome by its probability and adding up the results. Notice the connection to the mean. A plain mean weights every value equally, as if each had the same chance. Expected value weights each outcome by how likely it is. In fact, the expected value of a distribution is exactly its mean, the center of gravity of all the possible outcomes weighted by their probabilities.

The formula in symbols

In plain words: multiply each outcome by its probability, then add all those products together. That is the entire calculation, and it is exactly what the worked example below does.

The long-run average

The expected value is, in effect, the long-run average you would get if the random situation were repeated many times. Take a bet that pays you 10 dollars with probability 0.3 and costs you 2 dollars with probability 0.7. The expected value is 0.3 times 10, which is 3 dollars, minus 0.7 times 2, which is 1 dollar 40, giving 1 dollar 60 per play. No single play produces exactly 1 dollar 60, but over many plays your average settles near that number. Expected value smooths the randomness into one representative figure.

Why it guides decisions

Expected value is the basis for judging risky decisions because it weighs the entire range of outcomes by their probabilities, not just the best case or the single most likely one. A rational decision-maker looks at the whole distribution. This is why lotteries are poor bets. They have negative expected value, paying out far less on average than they cost. And it is why a casino, with a tiny edge on each bet, profits reliably over millions of bets.

Expected value weighs every outcome by its probability. A good decision considers the whole distribution, not just the hoped-for result.

The critical limitation

Here is a warning that runs through the rest of finance. Expected value alone can mislead you, because it ignores the spread and shape of outcomes. It treats a steady, modest gain the same as a wild gamble with the same average. A bet can have a positive expected value and still ruin you, if the path to that average includes a chance of a large, unrecoverable loss. Picture a wager that usually wins a little but occasionally wipes out everything you own. Its expected value might be positive, yet taking it over and over would eventually wipe you out. This is why expected value must always be paired with a measure of risk, the variance and the worst cases, which is where the next lessons go.

Why insurance makes sense

This limitation also explains insurance. For you, insurance has negative expected value, because the company charges a premium a bit higher than the average payout, which is how it profits. Yet buying it can still be perfectly rational, because it removes the small chance of a devastating, life-changing loss. You trade a slightly worse average for far greater safety. The mature use of expected value always looks at the distribution behind it, especially the downside, a theme that carries straight into variance, standard deviation, and portfolio risk.

Expected value tells you what to expect on average. To measure how far reality is likely to stray from that average, the risk, you need a measure of spread. The next lesson builds it, and explains why it squares the differences from the mean.