Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 14

Fig. 1.9 Illustration of total probability

This result is known as total probability because it finds the probability of A in
terms of conditional probabilities given B and given the complement of B .

Example 1.6.1 A device can operate under two kinds of conditions in time t :

Regular ; ð event AÞ

Nonregular

AÞ:

; ð event (1.60)

The corresponding probabilities are:

PfAg ¼ 0 : 85 ; PfAg ¼ 1 PfAg ¼ 0 : 15 : (1.61)

The probability that the device will fail under regular conditions in certain time t ,
is 0.1, in nonregular conditions it may fail with a probability of 0.7.
Find the total probability P { F } that the device will fail during the time t .

Solution The conditional probability that the device will fail, given A , is:

PfFjAg ¼ 0 : 1 : (1.62)

Similarly, the c o nditional probability that the device will fail under nonregular
conditions (given A ) is:

PfFjAg ¼ 0 : 7 : (1.63)

Using the formula for total probability ( 1.59 ), we find:

PfFg ¼ PfFjAgPfAgþPfFjAgPfAg ¼ 0 : 1 0 : 85 þ 0 : 7 0 : 15 ¼ 0 : 19 :

(1.64)

The results ( 1.58 ) and ( 1.59 ) can be generalized as discussed in the following.

Theorem of Total Probability Let B 1 , ..., B N be a set of mutually exclusive
events (i.e., B i \B j ¼ 0, for all i 6¼ j), and event A is the union of N mutually
exclusive events (A\B i ), i ¼ 1, ..., N.