Page 41 - Fundamentals of Probability and Statistics for Engineers

24 Fundamentals of Probability and Statistics for Engineers

The utility of this result rests with the fact that the probabilities in the sum
in Equation (2.27) are often more readily obtainable than the probability of A itself.
Example 2.11. Our interest is in determining the probability that a critical
level of peak flow rate is reached during storms in a storm-sewer system on the
basis of separate meteorological and hydrological measurements.
Let B i , i ˆ 1, 2, 3, be the different levels (low, medium, high) of precipitation
caused by a storm and let A j , j ˆ 1, 2, denote, respectively, critical and non-
critical levels of peak flow rate. Then probabilities P(B i ) can be estimated from
meteorological records and P A j jB i ) can be estimated from runoff analysis.
Since B 1 , B 2 , and B 3 constitute a set of mutually exclusive and exhaustive
events, the desired probability, P(A 1 ), can be found from

P A 1 †ˆ P A 1 jB 1 †P B 1 †‡ P A 1 jB 2 †P B 2 †‡ P A 1 jB 3 †P B 3 †:

Assume the following information is available:

P B 1 †ˆ 0:5; P B 2 †ˆ 0:3; P B 3 †ˆ 0:2;

and that P A j jB i ) are as shown in Table 2.2. The value of P(A 1 )is given by

P A 1 †ˆ 0 0:5†‡ 0:2 0:3†‡ 0:6 0:2†ˆ 0:18:
Let us observe that in Table 2.2, the sum of the probabilities in each column is
1.0 by virtue of the conservation of probability. There is, however, no such
requirement for the sum of each row.
A useful result generally referred to as Bayes’ theorem can be derived based
on the definition of conditional probability. Equation (2.24) permits us to write

P AB†ˆ P AjB†P B†

and

P BA†ˆ P BjA†P A†:

Since P AB) ˆ P BA), we have Theorem 2.2.

Table 2.2 Probabilities P A j jB i ), for Example 2.11
A j B i
B 1 B 2 B 3

A 1 0.0 0.2 0.6
A 2 1.0 0.8 0.4

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