Fundamentals of Probability and Statistics for Engineers

that

The probability of interest in part (b) is and this can be found using
Bayes’ theorem [Equation (2.28)]. It is given by:

It is worth mentioning that P(B) in this calculation is found by means of the
total probability theorem. H ence, Equation (2.29) is the one actually used here
in fi nding In fact, probability P(A) in Equation (2.28) is often more
conveniently found by using the total probability theorem.

Example 2.13.Problem: from Example 2.11, determine the probabil-
ity that a noncritical level of peak flow rate will be caused by a medium-level storm.
Answer: from Equations (2.28) and (2.29) we have

In closing, let us introduce the use of tree diagrams for dealing with more
complicated experiments with ‘limited memory’. Consider again Example 2.12

0.4

0.6

0.95

0.9

(^0). 0
5
0.^1
B
A B
A
Figure 2.7 Probabilities associated with a binary channel, for Example 2.12
26 Fundamentals of Probability and Statistics for Engineers
P…B†ˆP…BjA†P…A†‡P…BjA†P…A†ˆ 0 : 95 … 0 : 4 †‡ 0 : 1 … 0 : 6 †ˆ 0 : 44 :
P AjB),
P…AjB†ˆ
P…BjA†P…A†
P…B†

ˆ

0 : 95 … 0 : 4 †

0 : 44

ˆ 0 : 863 :

P AjB).

P B 2 jA 2 ),

P…B 2 jA 2 †ˆ

P…A 2 jB 2 †P…B 2 †

P…A 2 †

ˆ

P…A 2 jB 2 †P…B 2 †

P…A 2 jB 1 †P…B 1 †‡P…A 2 jB 2 †P…B 2 †‡P…A 2 jB 3 †P…B 3 †

ˆ

0 : 8 … 0 : 3 †

1 : 0 … 0 : 5 †‡ 0 : 8 … 0 : 3 †‡ 0 : 4 … 0 : 2 †

ˆ 0 : 293 :