Probability of employees traveling to work

[tzx9633]

Homework Statement

According to a survey, of those employees living more than 2 miles from work , 90% travel to work by car . Of the remaining employees, only 50% travel to work by car .

It's known that 75% of employees live more than 2 miles from work .

Find the probability of that an employee who travels to work by car more than 2 miles from work . The ans given is (0.9 x 0.75 ) / (0.75)(0.9) + (0.25)(0.5) = 0.844

Homework Equations

The Attempt at a Solution

But , I think it should be (0.9 x 0.75 ) = 0.675

Correct me if i am wrong .

 

[phinds]

Correct me if i am wrong .

I believe you are right and that the creator of the problem confused himself and did not ask what he thought he was asking.

 

[Ray Vickson]

Homework Statement

According to a survey, of those employees living more than 2 miles from work , 90% travel to work by car . Of the remaining employees, only 50% travel to work by car .

It's known that 75% of employees live more than 2 miles from work .

Find the probability of that an employee who travels to work by car more than 2 miles from work .The ans given is (0.9 x 0.75 ) / (0.75)(0.9) + (0.25)(0.5) = 0.844

Homework Equations

The Attempt at a Solution

But , I think it should be (0.9 x 0.75 ) = 0.675

Correct me if i am wrong .

I hope your instructor did not really ask you the highly ungrammatical question exactly as you wrote it. If the question had asked "Find the probability that an employee who travels by car lives more than 2 miles from work" then the instructor's solution would be correct. It would be asking for the conditional probability $P(G|C)$, where G is the event "lives Greater than 2 miles from work" and C is the event "travels by Car".

Also: I hope the instructor did not say that (0.9 x 0.75 ) / (0.75)(0.9) + (0.25)(0.5) = 0.844, because that would be wrong: you have
$$ \frac{(0.9 \times 0.75)}{(0.75)(0.9)} + (0.25)(0.5)$$
on the left, that that does not come anywhere near 0.844. However, writing (0.9)(0.75)/((0.75)(0.9) + (0.25)(0.5)) would give you a correct equation (except for computational roundoff errors). Basically a/b+c means $\frac{a}{b} + c$ when read using standard mathematical expression parsing rules; if we want $\frac{a}{b+c}$ we need to use parentheses, like this: a/(b+c).

 

[tzx9633]

"Find the probability that an employee who travels by car lives more than 2 miles from work" then the instructor's solution would be correct. It would be asking for the conditional probability P(G|C)P(G|C)P(G|C), where G is the event "lives Greater than 2 miles from work" and C is the event "travels by Car".

Why ? it's not stated earlier it's the conditional probability ...

IMO , if it's asking for conditional probability , it should be Find the probability that an employee who travels by car lives more than 2 miles from work given that the people travel to work by car

 

[phinds]

I believe you are right ...

OOPS. No I don't. I think you and I made the same mistake in reading the problem.

 

[tzx9633]

OOPS. No I don't. I think you and I made the same mistake in reading the problem.

Why ?

 

[phinds]

Why ?

Because in our computations we are leaving out the people who DO travel to work by car but do NOT live more than 2 miles away. We are ignoring them but they are part of the population that we need to include.

 

[Ray Vickson]

Why ? it's not stated earlier it's the conditional probability ...

IMO , if it's asking for conditional probability , it should be Find the probability that an employee who travels by car lives more than 2 miles from work given that the people travel to work by car

No, not necessarily: sometimes people say things like "find the probability that the person is a B if they are an A" instead of "find the (conditional) probability that a person is a B, given that they are an A". The first is a perfectly acceptable shortening of the second because it still includes an exact statement of what is assumed (in this case, A). Some people regard the word "conditional" as optional if the statement already has a "given ... " phrase--that is why I put the word "conditional" in parentheses in the second form.

Anyway, that all begs the question: is what YOU wrote exactly the same as what the instructor wrote? I hope not, as I already said.

 

[tzx9633]

No, not necessarily: sometimes people say things like "find the probability that the person is a B if they are an A"

Well , up to here , i can understand that it's conditional statement ...
But , in the original sentence ,
Find the probability of that an employee who travels to work by car more than 2 miles from work .
I can't understand why it's conditional statement , there's no the word 'if' in the sentence to show that it's conditional statement ... Can you explain further why it's conditional statement ?

 

[Ray Vickson]

Well , up to here , i can understand that it's conditional statement ...
But , in the original sentence ,
Find the probability of that an employee who travels to work by car more than 2 miles from work .
I can't understand why it's conditional statement , there's no the word 'if' in the sentence to show that it's conditional statement ... Can you explain further why it's conditional statement ?

I agree with you completely; that is why I called the question ungrammatical. Perhaps "meaningless" would have been a better term, followed by "ambiguous" as a close second. One can attempt to re-word the question so it make sense, but there will be several ways of doing that, leading to several different proper questions. I went with the one that a teacher would most likely want to express, but of course, that could be wrong. (I went with the one the I would have asked if I had set the problem.)

 

[tzx9633]

I agree with you completely; that is why I called the question ungrammatical. Perhaps "meaningless" would have been a better term, followed by "ambiguous" as a close second. One can attempt to re-word the question so it make sense, but there will be several ways of doing that, leading to several different proper questions. I went with the one that a teacher would most likely want to express, but of course, that could be wrong. (I went with the one the I would have asked if I had set the problem.)

It looks like I have to accept Find the probability of that an employee who travels to work by car more than 2 miles from work
as
$P(\text{lives >2 miles from work} | \text{drives to work}) $ without any strong /detailed explanation.