Expected Value

Start with the 1 in 1,000 shot at 500 dollars. Most plays win nothing, so if the ticket costs even a few dollars, you are almost certainly losing money that play. Expected value explains why that feels true across thousands of plays: multiply each outcome by how often it happens, add those up, and you get the one number that summarizes the whole game.

The mechanism is a weighted sum. For each possible outcome, multiply its payout by its probability. Add all those products together. That total is the expected payout. If the game costs money to play, subtract the cost at the end to get the expected value of playing. Every outcome contributes to the total in proportion to how likely it is, so rare outcomes barely move the number even if they pay huge amounts, and common outcomes dominate even if they pay little.

Expected value is not a prediction about one play. A single play of a game can only return one of its actual payouts, like 10 dollars or 0 dollars, never the blended number you calculated. The expected value only shows up as the average once you play many times and let the law of large numbers smooth out the randomness. This is the same idea behind a weighted grade average: each score counts in proportion to its weight, not its own value, and the final grade is a blend no single assignment actually produced.