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 and from work.

Decision Alternative	State of Nature
Expressway Open s1	Expressway Jammed s2
Queen City Avenue, d1	40	40
Expressway, d2	35	55

Based on their experience with traffic problems, Rona and Jerry agreed on a 0.05 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.

(a)

Use Bayes' theorem for probability revision to compute the probability of each weather condition and the conditional probability of the expressway open, s₁, or jammed, s₂,

given each weather condition. (Round the probability of each weather condition to three decimal places and the conditional probabilities to two decimal places.)

a)

Use Bayes' theorem for probability revision to compute the probability of each weather condition and the conditional probability of the expressway open,

s₁,

or jammed,

s₂,

given each weather condition. (Round the probability of each weather condition to three decimal places and the conditional probabilities to two decimal places.)

P(C)

P(O)

P(R)

P(s₁|C)

P(s₁|O)

P(s₁|R)

P(s₂|C)

P(s₂|O)

P(s₂|R)

(c)

What is the optimal decision strategy.

Check the weather and take the Queen City Avenue unless there is rain.

If it is raining, take the expressway.

Do not check the weather and take the expressway. Check the weather and take the expressway unless there is rain. If it is raining, take the Queen City Avenue.

Do not check the weather and take the Queen City Avenue.

What is the expected travel time (in min)? (Round your answer to one decimal place.)

(b) Show the decision tree for this problem. Decision Tree Description (i)