Conditional Probability - Faulty Plumbing

[newguy2]

This question has been driving me crazy.

A large industrial firm uses three 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 Lakeview. What is the probability that:

P(R) = 'Probability of being assigned to Ramada' = 20% = .20
P(S) = 'Probability of being assigned to Sheraton' = 50% = .50
P(L) = 'Probability of being assigned to Lakeview' = 30% = .30

P(F) = 'The probability of faulty plumbing' = ?

P(F | R) = 'Given room is at Ramada, prob. of faulty plumbing' = 5% = .05
P(F | S) = 'Given room is at Sheraton ...' = 4% = .04
P(F | L) = 'Given room is at Lakeview ...' = 8% = .08

Right?
So...:

A) What is the probability that a client will be assigned a room with faulty plumbing?

P(F) = P(R)P(F|R) + P(S)P(F|S) + P(L)P(F|L) = .20*.05 + .50*.04 + .30*.08 = 5.4% = .054
This makes sense...ok..
But...

B) What is the probability that a person with a room having faulty plumbing was assigned accommodations at Lakeview?

P(L | F) is what we are looking for, yes?
P(L | F) = 'Prob. of being assigned to LakeView, given room has faulty plumbing"
Right?

P(L | F) = P(L n F) / P(F) = P(F | L) P(F) / P(F) = P(F | L)...? This answer is not correct... how come?

P(L | F) = P(L n F) / P(F) = P(L) P(F) / P(F) = P(L)...? This answer is also not correct...

P(L | P(F|L)) = P(L n [F | L]) / P(F | L) = P(L)P(F | L) / P(F | L) = P(L) Still incorrect answer...

But this works...?

P(L | F) = P(L) P(F | L) / P(F) = correct answer?Please clarify all this for me.. What is happening.

 

[MarkFL]

Hello, and welcome to MHB! (Wave)

I've moved your post into its own thread...this way the discussion will be all about your question only, and you'll be more likely to get help too. :)

 

[newguy2]

P(L | F) = P(L n F) / P(F)

P(F | L) = P(F n L) / P(L)P(L n F) = P(L | F) P(F)
P(F n L) = P(F | L) P(L)

P(L n F) = P(F n L)

P(L n F) = P(F | L) P(L)
P(F n L) = P(L | F) P(F)

P(L | F) = P(L n F) / P(F) = P(F | L) P(F) / P(F) = P(F | L)...? This answer is not correct... how come?

Because P(L n F) != P(F | L) P(F)
it is P(L n F) = P(F | L) P(L)
or... P(L n F) = P(L | F) P(F)

P(L | F) = P(L n F) / P(F) = P(L) P(F) / P(F) = P(L)...? This answer is also not correct...

Because P(L n F) != P(L) P(F), ? How come? Independence thing?
so P(L n F) = P(L) + P(F) - P(L u F) ??

P(L | P(F|L)) = P(L n [F | L]) / P(F | L) = P(L)P(F | L) / P(F | L) = P(L) Still incorrect answer...

I get this is probably not even valid, this was just experimenting trying to see it gave the correct answer

But this works...?
P(L | F) = P(L) P(F | L) / P(F) = correct answer?
Please clarify all this for me.. What is happening.

I under stand why this works now anyways, because:
P(L n F) = P(F | L) P(L) = P(L) P(F | L)

 

[HallsofIvy]

This question has been driving me crazy.

A large industrial firm uses three 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 Lakeview. What is the probability that:

P(R) = 'Probability of being assigned to Ramada' = 20% = .20
P(S) = 'Probability of being assigned to Sheraton' = 50% = .50
P(L) = 'Probability of being assigned to Lakeview' = 30% = .30

P(F) = 'The probability of faulty plumbing' = ?

P(F | R) = 'Given room is at Ramada, prob. of faulty plumbing' = 5% = .05
P(F | S) = 'Given room is at Sheraton ...' = 4% = .04
P(F | L) = 'Given room is at Lakeview ...' = 8% = .08

Right?
So...:

A) What is the probability that a client will be assigned a room with faulty plumbing?r

P(F) = P(R)P(F|R) + P(S)P(F|S) + P(L)P(F|L) = .20*.05 + .50*.04 + .30*.08 = 5.4% = .054
This makes sense...ok..
But...

Imagine 1000 clients. 20% of them, 200, are assigned to Ramada, 50% of them, 500, are assigned to Sheraton, and 30%, 300, are assigned to Lakeview.

Of the 200 assigned to Ramada, 5%, 10, have faulty plumbing. Of the 500 assigned to Sheraton, 4%, 20, have faulty plumbing. Of the 300 assigned to Lakeview, 8%, 24, have faulty plumbing. That is a total of 10+ 20+ 24= 54 or 54/1000= 0.054, 5.4% of the clients, have faulty plumbing.

B) What is the probability that a person with a room having faulty plumbing was assigned accommodations at Lakeview?

Of the 54 people who had faulty plumbing, 24, so 24/54= 0.4444 or 44.44%, were assigned to Lakeview.

P(L | F) is what we are looking for, yes?
P(L | F) = 'Prob. of being assigned to LakeView, given room has faulty plumbing"
Right?

Yes, that is correct.

P(L | F) = P(L n F) / P(F) = P(F | L) P(F) / P(F) = P(F | L)...? This answer is not correct... how come?

P(L | F) = P(L n F) / P(F) = P(L) P(F) / P(F) = P(L)...? This answer is also not correct...

P(L | P(F|L)) = P(L n [F | L]) / P(F | L) = P(L)P(F | L) / P(F | L) = P(L) Still incorrect answer...

But this works...?

P(L | F) = P(L) P(F | L) / P(F) = correct answer?Please clarify all this for me.. What is happening.

You seem to be assuming. when you write, for example, "P(L n F)= P(L)P(F)", that "faulty plumbing" is independent of which motel a client is assigned to- and that is clearly not true.