You can classify pavements into two groups rigid and flexible If the pavement is

Question

You can classify pavements into two groups rigid and flexible If the pavement is rigid there is a 70% chance it is in good condition after 2 years, 20% chance it is in fair condition, and 10% it condition. If the pavement is flexible there is a 50% chance it is in good condition after 2 years, 30% chance it is in fair condition, and 20% poor condition. 30% of the pavements are rigid What is the probability that a randomly chose pavement will be in good condition after 2 years? What is the probability that a the pavement is rigid given it is in poor condition after 2 years?

Explanation / Answer

Let R shows the event that pavement is rigid and F shows the event that pavement is flexible. So we have

P(R) = 0.30, P(F) = 1 - P(R) = 0.70

Let G shows the event that pavemnet is in good condition, A shows the event that pavement is in fair condition and P shows the event that pavement is in Poor condition. So we have

P(G|R) = 0.70, P(A|R) = 0.20, P(P|R) = 0.10

P(G|F) = 0.50, P(A|F) = 0.30, P(P|A) = 0.20

By the law of total probability, the probability that pavement will be in good condition after 2 years is

P(G) = P(G|R)P(R) +P(G|F)P(F) = 0.70 * 0.30 + 0.50* 0.70 = 0.20 + 0.35 = 0.56

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By the law of total probability, the probability that pavement will be in poor condition after 2 years is

P(P) = P(P|R)P(R) +P(P|F)P(F) = 0.10 * 0.30 + 0.20* 0.70 = 0.17

So by the Baye's theorem the probability that the pavement is rigid given it is in poor condition after 2 years is

P(R|P) = [P(P|R)P(R)] / P(P) = 0.03 / 0.17 = 0.1765

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