Page 40 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 23

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

P A 1 A 2 ... A n †ˆ P A 1 †P A 2 jA 1 †P A 3 jA 1 A 2 † ... P A n jA 1 A 2 ... A n 1 †: 2:26†

where P(A i ) > 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: theorem of total probability. Suppose that events B 1 , B 2 , .. . , and
B n are mutually exclusive and exhaustive (i.e. S ˆ B 1 ‡ B 2 ‡ ‡ B n ). Then,
for an arbitrary event A,
P A†ˆ P AjB 1 †P B 1 †‡ P AjB 2 †P B 2 † ‡ ‡ P AjB n †P B n †
n
X 2:27†
ˆ P AjB j †P B j †:
jˆ1
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 ,..., AB n (i.e.
A ˆ AB 1 ‡ AB 2 ‡ ‡ AB n ). Hence,
P A†ˆ P AB 1 †‡ P AB 2 † ‡ ‡ P AB n †;
which gives Equation (2.27) on application of the definition of conditional
probability.
AB 1 A AB 3 AB 5

B 3

B 1
B 5
S

B 4
B 2

AB 2 AB 4
Figure 2.6 Venn diagram associated with total probability

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