Question

In: Statistics and Probability

Please answer the problem below for all parts. Please show all work and write clearly. Thanks....

Please answer the problem below for all parts. Please show all work and write clearly. Thanks.

(Apply total probability and Bayes’ rules)

A large industrial firm uses three local motels to provide overnight accommodations for its clients. From past experience it is known that 22% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton, and 28% at the Lakeview Motor Lodge. If the plumbing is faulty in 5% of the rooms at the Ramada Inn, in 4% of the rooms at the Sheraton, and in 8% of the rooms at the Lakeview Motor Lodge, 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 Lakeview Motor Lodge?

Solutions

Expert Solution

Let

R: a client is assigned accommodation at Ramada Inn

Given, P(R) = 0.22

S: a client is assigned accommodation at Sheraton

Given, P(S) = 0.50

L: a client is assigned accommodation at Lakeview Motor Lodge

Given, P(L) = 0.28

P(F|R): Probability of faulty plumbing rooms in Ramada = 0.05

P(F|S): Probability of faulty plumbing rooms in Sheraton = 0.04

P(F|L): Probability of faulty plumbing rooms in Lakeview = 0.08

Answer a)

Now, we need to find probability that client is assigned with faulty plumbing room P(F)

Using theorem of total probability, we get:

P(F) = P(R)*P(F|R) + P(S)*P(F|S) + P(L)*P(F|L) = 0.22*0.05 + 0.50*0.04 + 0.28*0.08

P(F) = 0.0534

Answer b)

Now, we need to find probability that faulty plumbing room was assigned at Lakeview Motor Lodge P(L|F)

Using Bayes' Theorem, we get:

P(L|F) = [P(F|L)*P(L)]/[P(R)*P(F|R) + P(S)*P(F|S) + P(L)*P(F|L)]

P(L|F) = 0.28*0.08/(0.22*0.05 + 0.50*0.04 + 0.28*0.08)

P(L|F) = 0.28*0.08/0.0534

P(L|F) = 0.4195


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