Q2 Probability [18 marks]

You have a model that has several events: Alarm = the fire alarm in your apartment sounds, Fire = there was a fire in your apartment, and Tampering = your smoke detector was tampered with. Fire affects Alarm (a fire increases the probability of the alarm sounding) and Tampering also affects Alarm (if someone tampered with your smoke detector, your alarm is more likely to sound when there is no fire, and less likely to sound when there is a fire). Smoke (you can see smoke in your apartment), Evacuation (your apartment building is evacuated), and Report (the local newspaper writes a report about the evacuation of your apartment).

P (Tampering)= T P (Tampering)= F

.02 .98

P (Fire=T) P(Fire ) = F

.01 .99

P(Alarm = T| Tampering , Fire ) P(Alarm = F| Tampering , Fire )

Tampering = T, Fire = T .5 .5

Tampering = T, Fire = F .85 .15

Tampering = F, Fire = T .99 .01

Tampering = F, Fire = F 0 1

The probability of smoke when there is fire is 0.95 and the probability of smoke when there is no fire is 0.05. When your apartment building has a fire alarm, there is a 0.9 probability that there will be an evacuation, but there is never an evacuation when there is no fire alarm. If there is an evacuation, there is a 0.7 probability that the newspaper will write a report on it, and if there is no evacuation there is a 0.9 probability that the newspaper will not report it. You could use Netica to solve this question and present the network setup.

a.What is the marginal probability that your smoke detector has been tampered with?

b.What is the marginal probability that there will be a news report tomorrow?

c.Let's assume that you have observed that there is smoke in your apartment. What is the posterior probability that there will be a news report tomorrow?

d.Let's assume that you have observed that there was no fire, and that there was a news report about your apartment. What is the posterior probability that your smoke detector has been tampered with?

e.Let's assume that you have observed that there is no smoke in your apartment. What is the posterior probability that your smoke detector has been tampered with? What conditional independence property could help you here?

f.Let's assume that you have observed that there has been a news report about your apartment, and there is no smoke in your apartment. What is the posterior probability that your smoke detector has been tampered with? Given that the news report was observed, why does observing the absence of smoke affect your belief of whether or not your smoke alarm was tampered with?

g.Let's assume that you have observed that there was no fire, that there was a news report about your apartment, and that there is smoke in your apartment. What is the posterior probability that your smoke detector has been tampered with? How does observing whether or not there is smoke affect your belief of whether or not your smoke detector has been tampered with?