3.5 The Theorem of Total Probability

Theorem 3.5.1

Let { A _i, i = 1, , n} be a partition. Then for any event B

Proof The theorem follows immediately from Theorem 3.3.3 since P( B ? A _i) = P( B A _i) P( A _i) for every i.

The importance of the theorem of total probability lies in the fact that we can often easily evaluate the probability of some event conditional on the occurrence of some other events. The theorem then enables us to evaluate the unconditional probability.

Example Suppose that fires in a compartment can be classified into three types with the given probabilities:

1. F ₁: Smouldering fire, P( F ₁) = 0 .2.

2. F ₂: Flaming fire, P( F ₂) = 0 .4.

3. F ₃: Flashover fire, P( F ₃) = 0 .4.

We are interested to find out the probability of death of an occupant of the compartment. Let us denote the event of death by D and suppose that the probability of death, conditional on the type of fire, is given by the following figures: P( D F ₁) = 0 .01, P( D F ₂) = 0 .3 and P( D F ₃) = 0 .699. Then, using the theorem...