Calculating the Probability of Faulty Plumbing in Hotel Rooms: A Case Study

[Roohul Amin]

Dear all Please help in solving the following problem.
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 sheeraton and 30% 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 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 accommodation at the Lakeview Motor Lodge?
I have come with the following solution.
Let L: a client is assigned accommodation at Lakeview Motor Lodge
R: a client is assigned accommodation at Ramada Inn
S: a client is assigned accommodation at Sheraton
P(Lf): Probability of faulty plumbing rooms in Lakeview=0.08
P(Rf): Probability of faulty plumbing rooms in Ramada=0.05
P(Sf): Probability of faulty plumbing rooms in Sheraton=0.04
Now
a) P(client assigned with faulty plumbing room)=P(Lf/L)+P(Rf/R)+P(Sf/S)
=0.3*0.08+0.2*0.05+0.5*0.04
b) P(faulty plumbing room in Lakeview)=P(L/Lf)
=0.08*0.3

Needs help

 

[Opalg]

Dear all Please help in solving the following problem.
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 sheeraton and 30% 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 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 accommodation at the Lakeview Motor Lodge?
I have come with the following solution.
Let L: a client is assigned accommodation at Lakeview Motor Lodge
R: a client is assigned accommodation at Ramada Inn
S: a client is assigned accommodation at Sheraton
P(Lf): Probability of faulty plumbing rooms in Lakeview=0.08
P(Rf): Probability of faulty plumbing rooms in Ramada=0.05
P(Sf): Probability of faulty plumbing rooms in Sheraton=0.04
Now
a) P(client assigned with faulty plumbing room)=P(Lf/L)+P(Rf/R)+P(Sf/S)
=0.3*0.08+0.2*0.05+0.5*0.04
b) P(faulty plumbing room in Lakeview)=P(L/Lf)
=0.08*0.3

Needs help

Your answer for a) is 0.3*0.08+0.2*0.05+0.5*0.04 = 0.054, which is correct.

For b), you need to know what proportion of the clients with faulty plumbing were staying at Lakeview. In other words, you need to know $$\dfrac{\text{P(faulty plumbing room in Lakeview)}}{\text{P(client with faulty plumbing room)}} = \dfrac{0.024}{0.054}.$$

 

[HallsofIvy]

Imagine 1000 clients. 20%, 200, are assigned the Ramada Inn, 50%, 500, the Sheraton, and 30%, 300, the Motor Lodge.

Of the 200 assigned to the Ramada Inn, 5%, 10, have faulty plumbing. Of the 500 assigned to the Sheraton, 4%, 20, have faulty plumbing, and of the 300 assigned to the Motor Lodge, 8%, 24, have faulty plumbing. That is a total of 10+ 20+ 24= 54, out of 1000 who have faulty plumbing- 54/1000= 0.054 or 5.4% are assigned a room with faulty plumbing. Of those 54, 24 were assigned to the Motor Inn. Given that a person had faulty plumbing, the probability they were assigned to the Motor Inn is 24/54= 0.444 or 44.4%.