Defining Expected Value
Expected value (EV) represents the long-term average outcome of a random variable. It's a fundamental concept in probability and statistics, often used to predict the average result if an event were to be repeated many times. Unlike a single specific outcome, EV provides a theoretical average, weighted by the probability of each potential outcome.
Calculating Expected Value
To calculate the expected value, you multiply each possible outcome by its respective probability and then sum these products. Mathematically, for a discrete random variable X with outcomes x1, x2, ..., xn and probabilities p1, p2, ..., pn, the Expected Value E(X) = (x1 * p1) + (x2 * p2) + ... + (xn * pn). This formula ensures that outcomes with higher probabilities contribute more to the overall average.
A Practical Example
Consider a simple game where you flip a fair coin. If it lands on heads, you win $10. If it lands on tails, you lose $5. The probability of heads is 0.5, and tails is 0.5. The expected value for playing this game is E(X) = ($10 * 0.5) + (-$5 * 0.5) = $5 + (-$2.50) = $2.50. This means that, on average, you would expect to win $2.50 per game if you played many times.
Importance and Applications
Expected value is crucial for making informed decisions under uncertainty across various fields. In finance, it helps assess investment risks and returns. In insurance, companies use it to set premiums. In gambling, it determines the fairness or profitability of a game. It provides a quantitative measure to evaluate the potential benefits or costs of different choices over the long run.