Page 36 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 19
1
where B 1 , B 2 , and B 3 are mutually exclusive, each occurring with probability .
4
It is easy to calculate the following:
1
P A 1 †ˆ P B 1 [ B 2 †ˆ P B 1 †‡ P B 2 †ˆ ;
2
1
P A 2 †ˆ P A 3 †ˆ ;
2
1
P A 1 A 2 †ˆ P‰ B 1 [ B 2 †\ B 1 [ B 3 †Š ˆ P B 1 †ˆ ;
4
1
P A 1 A 3 †ˆ P A 2 A 3 †ˆ ;
4
P A 1 A 2 A 3 †ˆ P‰ B 1 [ B 2 †\ B 1 [ B 3 †\ B 2 [ B 3 †Š ˆ P ;† ˆ 0:

We see that Equation (2.17) is satisfied for every j and k in this case, but
Equation (2.18) is not. In other words, events A 1 , A 2 , and A 3 are pairwise
independent but they are not mutually independent.
In general, therefore, we have Definition 2.2 for mutual independence of
n events.

Definition 2.2. Events A 1 , A 2 ,..., A n are mutually independent if and only if,

with k 1 , k 2 ,..., k m being any set of integers such that 1 k 1 < k 2 ...< k m n
and m ˆ 2, 3, . . . , n,
†: 2:19†
P A k 1 A k 2 ... A k m †ˆ P A k 1 †P A k 2 † ... P A k m
n
n
The total number of equations defined by Equation (2.19) is 2 1.
Example 2.8. Problem: a system consisting of five components is in working
order only when each component is functioning (‘good’). Let S i , i ˆ 1, ..., 5, be
the event that the ith component is good and assume P(S i ) ˆ p i . What is the
probability q that the system fails?
Answer: assuming that the five components perform in an independent
manner, it is easier to determine q through finding the probability of system
success p. We have from the statement of the problem
p ˆ P S 1 S 2 S 3 S 4 S 5 †:

Equation (2.19) thus gives, due to mutual independence of S 1 , S 2 ,..., S 5 ,

p ˆ P S 1 †P S 2 † ... P S 5 †ˆ p 1 p 2 p 3 p 4 p 5 : 2:20†
Hence,
q ˆ 1 p ˆ 1 p 1 p 2 p 3 p 4 p 5 : 2:21†

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