Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 22

From ( 1.65 ), we obtain:

PfAg¼PfA 1 gPfAjA 1 gþPfA 2 gPfAjA 2 gþPfA 3 gPfAjA 3 g

0 : 5 n 1 þ

1

3 n 2 þ 0 : 5 n 3
n

¼ : (1.107)

Exercise E.1.7 A given system has two units. In order for the system to be in the
operational mode, each unit must be in the operational mode, as well. The failures
of units are independent. The probabilities of failure - free operations for units 1
and 2 are known and denoted as p 1 and p 2 , respectively.
Find the probability that the second unit fail, if it is known that system failed.
(The first unit worked).

Answer We define the following events:

A 1 ¼f both units work g;
A 2 ¼f the first unit failed and the second unit works g;
A 3 ¼f the first unit works and second unit failed g;
A 4 ¼f both units failed g : (1.108)

The corresponding probabilities are:

PfA 1 g¼p 1 p 2 ;
PfA 2 g¼ð 1 p 1 Þp 2 ;
PfA 3 g¼p 1 ð 1 p 2 Þ;
PfA 4 g¼ð 1 p 1 Þð 1 p 2 Þ: (1.109)

The system failed if any or both units failed.
Defining event A as:

A ¼f the system failed g; (1.110)

we write the corresponding conditional probabilities:

PfAjA 1 g¼ 0 ð If both units work the system cannot fail Þ

PAjAf ¼ PAjAf ¼ PAjAf ¼ 1 The system fails if any

;

ð ; or both units fail Þ:

(1.111)

Denote the event that the second unit fails, if we know that the system failed, as
( A 3 | A ).