Probability Trees and Conditional Expectations

Conditional Expected Values, the expected value of a random variable X given an event or scenario S denoted E(X|S). Suppose the random variable X can take on any one of n distinct outcomes $X_1, X2,\dots,X_n$ (these outcomes form a set of mutually exclusive and exhaustive events).

The expected value of X conditional on S is the first outcome, $X_1$, times the probability of the first outcome given S, $P(X_1|S)$, plus the second outcome, $X_2$, times the probability of the second outcome given S, $P(X_2|S)$, and so forth, as follows:

Total Probability Rule for Expected Value

The formula follows:

where $S^C$ is the “complement of S,” which means event or scenario “S” does not occur.

where $S_1,S_2,\dots,S_n$ are mutually exclusive and exhaustive scenarios or events.