A large industrial firm uses 3 local motels to provide overnight accommodations

Question

A large industrial firm uses 3 local motels to provide overnight accommodations for its clients. From past experience it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton, and 30% at the Holiday Inn. It is found (a given condition) that plumbing is faulty at 5% of the rooms at the Ramada Inn, 4% of the rooms at the Sheraton, and 8% of the rooms at the Holiday Inn. What is the probability that:  

A) A client will be assigned a room with faulty plumbing.

B) A person with a room having faulty plumbing was assigned accommodations at the Holiday Inn

Explanation / Answer

Let R shows the event that the clients are assigned rooms at the Ramada Inn, S shows the event that the clients are assigned rooms at the Sheraton and H shows the event that the clients are assigned rooms at the Holiday Inn. Let P shows the event that plumbing is faulty. Here we have following probabailites:

P(R) = 0.20, P(S) = 0.50, P(H) = 0.30

And following conditional probabailties:

P(P|R) = 0.05, P(P|S) = 0.04, P(P|H) = 0.08

(A)

By the law of total probabailty the probability that a client will be assigned a room with faulty plumbing is

P(P) = P(P|R)P(R) +  P(P|S)P(S) +  P( P|H)P(H) = 0.05 * 0.20 + 0.04*0.50+ 0.08*0.30 = 0.01 + 0.02 + 0.024 = 0.054

(B)

Here we need to calcuate the probabaility P(H|P) so by bayes theorem we have

P(H|P) = [P(P|H)P(H) ] / P(P) = 0.024 / 0.054 = 0.4444

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