In my friend’s town (not Vancouver), it\'s rainy one third of the days. Given th

Question

In my friend’s town (not Vancouver), it's rainy one third of the days. Given that it is rainy, there will be heavy traffic with probability 1/2, and given that it is not rainy, there will be heavy traffic with probability 1/4. If it's rainy and there is heavy traffic, I arrive late for work with probability 1/2. On the other hand, the probability of being late is reduced to 1/8 if it is not rainy and there is no heavy traffic. In other situations (rainy and no traffic, not rainy and traffic) the probability of being late is 0.25. You pick a random day.

a) What is the probability that it's not raining and there is heavy traffic and I am not late? Use both tree diagram and application of chain rule to show your result.

b) What is the probability that I am late? Use both tree diagram and the law of total probability to show your result.

c) Given that I arrived late at work, what is the probability that it rained that day?

Explanation / Answer

given that probability of rainy with heavy traffic = 1/2

probability of not rainy with traffic = 1/4

probability of arriving late due to traffic = 1/2

probability of arriving late without traffic = 1/8

a)

probability that its not raining and there is heavy traffic and im not late = (2/3)*(1/4)*(1-0.25)

= 0.125

b)

probability of being late = (1/3)(1/2)(1/2) + (1/3)(1/2)(0.25) + (2/3)(1/4)(0.25) + (2/3)(3/4)(1/8)

= 0.229

c)

probability (rainy day + arrived late) = ((1/3)*(1/2)*(1/2) + (1/3)*(1/2)*(0.25)) / (0.229) = 0.5454

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