In my town, it\'s rainy one third of the days. Given that it is rainy, there wil

Question

In my town, 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. What is the probability that it's not raining and there is heavy traffic and I am not late? What is the probability that I am late? Given that I arrived late at work, what is the probability that it rained that day?

Explanation / Answer

Probability that its not raining and there is heavy traffic and I am not late = (2/3)x(1/4)x(1-0.25) = 0.125

P(being late) = (1/3)x(1/2)x(1/2)+(1/3)x(1/2)x0.25 + (2/3)x(1/4)x0.25 + (2/3)x(3/4)x(1/8) = 0.2292

P(rainy day | arrived late) = ((1/3)x(1/2)x(1/2)+(1/3)x(1/2)x0.25))/0.2292 = 0.5454

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