Let be a partition of the sample space, and assume for all that . Then for any r.v. , the expectation can be expressed as

Expressing LOTP using LOTE

Consider any event , and let be an Indicator variable that equals 1 if occurs, and 0 otherwise. Using the same from above, we can use LOTE to express the expectation of as

Remember that simplifies to . Now we’re finding the probability that , given , which is equivalent to finding the probability that occurs, given .

We’ve just rediscovered LOTP using LOTE. In a sense, LOTP is a special case of LOTE when the random variable we’re finding the expectation of is an indicator variable that represents that event.