Question

2. (20 Points) Bayesian networks are used on the Web sites of high-technology manufactures to allow customers to quick diagnose problems with products. An oversimplified example is presented here. A printer manufacturer obtained the following probabilities from a database of test results. Printer failures are associated with three types of problems: hardware, software, and other (such as connecters), with probabilities 0.1, 0.6, and 0.3, respectively. The probability of a printer failure given a hardware problem is 0.9, given a software problem is 0.2, and given any other type of problem is 0.5. If a customer enters the manufacturer's Web site diagnose a printer failure, what is the most likely cause of the problem?

Answer 1

P(Hardware problem) = 0.1

P(Software problem) = 0.6

P(Other problem) = 0.3

P(Printer failure | Hardware problem) = 0.9

P(Printer failure | Software problem) = 0.2

P(Printer failure | Other problem) = 0.5

In order to determine the most likely cause of printer failure ,we need to determine

i) P( Hardware problem | Printer failure)

Using Bayes theorem, this is equal to

(0.9*0.1) / [ (0.9*0.1)+(0.2* 0.6) + (0.5*0.3) ]

= 0.09 / (0.09 + 0.12+0.15)

= 0.09 / 0.36

=9/36

ii) P( Software problem | Printer failure)

Using Bayes theorem, this is equal to

(0.2*0.6) / [ (0.9*0.1)+(0.2* 0.6) + (0.5*0.3) ]

= 0.12 / (0.09 + 0.12+0.15)

= 0.12 / 0.36

=12/36

iii) P( Other problem | Printer failure)

Using Bayes theorem, this is equal to

(0.5*0.3) / [ (0.9*0.1)+(0.2* 0.6) + (0.5*0.3) ]

= 0.15 / (0.09 + 0.12+0.15)

= 0.15 / 0.36

=15/36

Thus,we can see that the most likely cause in other problem