To save on expenses, Rona and Jerry agreed to form a carpool for traveling to an

Question

To save on expenses, Rona and Jerry agreed to form a carpool for traveling to and from work. Rona preferred to use the somewhat longer but more consistent Queen City Avenue. Although Jerry preferred the quicker expressway, he agreed with Rona that they should take Queen City Avenue if the expressway had a traffic jam. The following payoff table provides the one-way time estimate in minutes for traveling to or from work: Based on their experience with traffic problems, Rona and Jerry agreed on a 0.15 probability that the expressway would be jammed. In addition, they agreed that weather seemed to affect the traffic conditions on the expressway. Let C = clear O = overcast R = rain The following conditional probabilities apply: P(C|s_1) = 0.8 P(C|s_2) = 0.1 P(O|s_1) = 0.2 P(O|s_2) = 0.3 P(R|s_1) = 0.0 P(R|s_2) = 0.6 (a) Use Bayes' theorem for probability revision to compute the probability of each weather condition and the conditional probability of the expressway open, vi or jammed given each weather condition. (b) Show the decision tree for this problem. (c) What is the optimal decision strategy, and what is the expected travel time?

Explanation / Answer

Here,

p(s2)=0.15 , p(s1)=0.85

p(C|s1) = 0.8 , p(C|s2) = 0.1

p(O|s1) = 0.2 , p(O|s2) = 0.3

p(R|s1) = 0.0 , p(R|s2) = 0.6

1) P(express is open | whether is clear)

= P(s1|C) == P(s1) * P(C|s1) / P(s1) * P(C|s1) + P(s2) * P(C|s2)

= P(s1|C) = 0.68 / 0.695

= P(s1|C) = 0.978

2) P(express is open | whether is Overcast)

= P(s1|O) == P(s1) * P(O|s1) / P(s1) * P(O|s1) + P(s2) * P(O|s2)

= P(s1|O) = 0.17 / 0.215

= P(s1|O) = 0.791

3) P(express is open | whether is rainy)

= P(s1|R) == P(s1) * P(R|s1) / P(s1) * P(R|s1) + P(s2) * P(R|s2)

= P(s1|R) = 0.0 / 0.09

= P(s1|R) = 0

4) P(express is jammed | whether is clear)

= P(s2|C) == P(s2) * P(C|s2) / P(s1) * P(C|s1) + P(s2) * P(C|s2)

= P(s2|C) = 0.015 / 0.695

= P(s2|C) = 0.022

5) P(express is jammed | whether is Overcast)

= P(s2|O) == P(s2) * P(O|s2) / P(s1) * P(O|s1) + P(s2) * P(O|s2)

= P(s2|O) = 0.045 / 0.215

= P(s2|O) = 0.209

6) P(express is jammed | whether is rainy)

= P(s2|R) == P(s2) * P(R|s2) / P(s1) * P(R|s1) + P(s2) * P(R|s2)

= P(s2|R) = 0.09 / 0.09

= P(s1|R) = 1

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