Fundamentals of Probability and Statistics for Engineers

Equation (2.25) is seen to be useful for finding joint probabilities. Its exten-
sion to more than two events has the form

where P(Ai) > 0 for all i. This can be verified by successive applications of
Equation (2.24).
In another direction, let us state a useful theorem relating the probability of
an event to conditional probabilities.

Theorem 2. 1: t heorem of t ot a l probabilit y. Suppose that events B 1 ,B 2 ,..., and
Bn are mutually exclusive and exhaustive (i.e. S B 1 B 2 Bn). Then,
for an arbitrary event A,

Proof of Theorem 2.1:referring to the Venn diagram in Figure 2.6, we can
clearly write A as the union of mutually exclusive events AB 1 ,AB 2 ,...,ABn (i.e.
). H ence,

which gives Equation (2.27) on application of the definition of conditional
probability.

AB 1

B 1

B 2

B 3

B 5

B 4

AB 3

AB 2 AB 4

AB 5

S

A

Figure 2.6 Venn diagram associated with total probability

Basic Probability Concepts 23

P…A 1 A 2 ...An†ˆP…A 1 †P…A 2 jA 1 †P…A 3 jA 1 A 2 †...P…AnjA 1 A 2 ...An 1 †: … 2 : 26 †

ˆ‡ ‡‡

P…A†ˆP…AjB 1 †P…B 1 †‡P…AjB 2 †P…B 2 †‡‡P…AjBn†P…Bn†

ˆ

Xn

jˆ 1

P…AjBj†P…Bj†:

… 2 : 27 †

AˆAB 1 ‡AB 2 ‡‡ABn

P…A†ˆP…AB 1 †‡P…AB 2 †‡‡P…ABn†;