Decision Making with Bayesian Networks The following figure shows a simple Bayes

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Decision Making with Bayesian Networks The following figure shows a simple Bayesian network to help you decide where or not to speed on the highway on your way to your parent's home for Thanksgiving dinner. Each node in the network represents a random Boolean variable. The random variable Cop indicates whether there is a highway patrol cop present on the freeway; SeeCop indicates whether you have detected a cop or not. The variable SlowTraffic indicates whether traffic is moving slower than the posted speed limits and Speed indicates whether or not you are speeding. Ticket is true when you get a ticket, and OnTime indicates that you made it on-time for the festivities. You are given the following probabilities P(Cop = true)-01, P(Speed = true) = 0.25, PLSeeCop = true | Cop = true) = 0.6, P(SeeCop = true Cop = false) = 0.0 rSlowTraffic = true | Cop = true) = 0.8, P(SlowTraffic = true | Cop = false) = 0.3, PL Ticket = true | Cop, Speed) = | 0.5 if Cop, Speed are both true; otherwise 0 0.9 0.5 0.3 0.1 if Ticket is true if Ticket is false, Speed is true, SlowTraffic is false if Ticket is false, Speed is true, SlowTraffic is true if Ticket is false, Speed is false, SlowTraffic is false if Ticket is false, Speed is false, SlowTraffic is true prOnTime = true, Ticket, Speed, SlowTraffic) = Cop Speed SeeCop Ticket OnTime SlowTraffic For each of scenario si below, SI: You don't detect a cop and you speed S2: You don't detect a cop and you do not speed · s3: You don't detect a cop, but traffic is slow, and you speed S4: You don't detect a cop, but traffic is slow, and you do not speed (1) Compute the probability of receiving a ticket for each scenario s (2) Compute the probability of arriving on time for each scenario (3) Suppose the cost of a speeding ticket is S100, and you lose a $10 bet to your brother if you arrive late. Using your answers from questions (2-3), compute the expected utility in each scenario s (4) Now determine whether you should speed or not in these two scenarios (A) a cop is not detected, and (B) a cop is not detected, but traffic is slow

Explanation / Answer

The scenario is that the cop is not detected and we speed

1.

P(SeeCop=false) = P(SeeCop=F | Cop=F). P(Cop=F) + P(Seecop=F | Cop=T). P(Cop=T)

= 1*0.9 + 0.4*0.1 = 0.94

We want P(Ticket = T | SeeCop=F,Speed=T)

P(Cop=T | SeeCop=F) = P(SeeCop=F | Cop=T). P(Cop=T) / P(SeeCop=F)

= 0.4*0.1/0.94 = 0.043

So, P(Ticket = T | SeeCop = F, Speed=T) = P(Ticket = T | Cop=T, Speed = T). P(Cop=T | SeeCop=F) = 0.5*0.043 = 0.0215

2. On time arrival probability depends on whether ticket is given or not, speed and slowtraffic

When Ticket is true, P(Ontime) = 0

When Ticket = F and SlowTraffic=F,

Prob = 0.9*P(Ticket=F and Slowtraffic=F | Cop=T,Speed=T,SeeCop=F) . P(Cop, Speed=T | SeeCop = F)= 0.9*0.5*0.2*0.043*0.25 = 0.0009675

When Ticket = F and Slow Traffic = T

Prob =0.5*P(Ticket=F and Slowtraffic= | Cop=T,Speed=T,SeeCop=F) . P(Cop, Speed=T | SeeCop = F)= 0.5*0.5*0.8*0.043*0.25 = 0.00215

So, required probability = 0+ 0.0009675 + 0.00215 = 0.0031175

3.

Expected utility = 0.0215*(-100)+10*0.0031175 = -2.11

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