Definition
Law of Total Probability
If only the conditional probabilities and the probabilities of the conditional event are known, then the total probability of
results from: For event
to occur, it must happen either with event or without event . There are no other possibilities. Partition of the sample space (general case): Given a set of events
that form a partition of the sample space . A partition must satisfy two conditions:
- Pairwise disjoint: The events do not overlap, i.e.:
. - Collectively Exhaustive: The events cover the entire sample space, i.e.
. If these conditions are met, the probability of any event
is the sum of its conditional probabilities across all the partitions, weighted by the probability of each partition: