Theorem. If the event A is known to occur together with one of the events forming a complete group of mutually exclusive events, then the probability of the event A can be found from the formula
or
This formula is known as the total probability formula.
Proof. According to the condition the event A can occur together with one of the mutually exclusive events . In other words the event A can be represented as a union of the events
, i.e.
Using the probability addition rule, we obtain
.
Now, using the probability multiplication rule, we get
The theorem has been proved.
Let us now assume that the event A can occur together with one of the events forming a complete group of mutually exclusive events. Since we do not know which one of these events occurs they are called hypotheses.
