Expected value
Probability times payoff, summed across outcomes.
The honest way to compare gambles. List the possible outcomes, attach a probability to each, multiply by the payoff, sum. The result is the expected value of taking the bet.
A 50% chance of winning $200 has the same expected value as a 100% chance of winning $100. A rational decision-maker should be indifferent (ignoring risk aversion). Most people aren't, but that's about psychology, not math.
The practical use is to override gut intuition with explicit numbers when stakes are big. Most bad bets that look good and most good bets that look bad reveal themselves when you write out the expected-value calculation honestly.
Caveat: expected value can be a trap when the downside includes ruin (see [ruin-absorbing-barriers]). A 99% chance of $1M and a 1% chance of "the game is over" has fine expected value and is a terrible bet to take.
Examples in the wild
Pricing decisions often improve when you do the EV calculation explicitly: probability of acceptance × deal size, summed across pricing options. The intuitive answer is usually wrong by 20-30%.
Expected value is the foundational lens of professional investing, but it's badly misused. The probabilities aren't really known. The honest version always includes a margin of safety.
Lottery tickets have terrible EV. Insurance also has negative EV (the insurer needs to make money). Both can be rational if the downside without them is catastrophic.
Expected value is one of the mental models we apply through real cases inside the Pareto MBA — a part-time program for professionals who want to think clearly about business.