Expected value is the probability-weighted mean of a discrete random variable, calculated as , and interpreted as the long-run average outcome over many repetitions in the same context. It may not be an outcome that can occur in a single trial, and its units match the variable’s units; applications include assessing fair games, risks, and typical financial or statistical outcomes for finite distributions, without extending to continuous or advanced theoretical models.
Expected value is a probability-weighted average. It tells you the long-run average result if the same situation is repeated many times.
For a discrete random variable ,
This means: multiply each possible outcome by its probability, then add the products.
Example: A game costs $3 to play. The prize distribution is:
\vert Prize \vert Probability \vert \vert --- \vert ---: \vert \vert 00.50$40.40$200.10$ \vert
Let represent the prize money won.
Step 1: Multiply each outcome by its probability.
Step 2: Add the products.
The expected prize is 3.60$3.60 in one game; the possible prizes are only 0$4$20$.
Over many plays, however, the average prize won per game would approach 3.60$3$, the expected net result is
So, in the long run, a player would gain an average of 0.60$ per game, even though individual games may result in a loss or a much larger win.
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