The distance between Urbania and Metropolis is 30 miles, with numbered

Question

The distance between Urbania and Metropolis is 30 miles, with numbered markers indicating each mile out of Urbania. If a person starts at a random location on the road between the two cities and drives exactly five miles in the direction of Metropolis—without passing the city—what is the probability that this person ends within five miles of the sign indicating 25 miles from Urbania?

(A) \(16 \frac{1}{6}%\)
(B) \(20%\)
(C) \(33 \frac{1}{3}%\)
(D) \(40%\)
(E) \(80%\)

(A)  16 \frac{1}{6}% 
(B)  20% 
(C)  33 \frac{1}{3}% 
(D)  40% 
(E)  80%

chetan2u · Answer

Probability is almost always between 0 and 1, both inclusive. So I believe we are looking for likelihood in %.

If the person travels 5 miles towards M, he can never be within 5 miles from U.

So total event = 5 to 30 = 25
Event of being within 5 miles from 25 mile marker = 20 to 30 = 10

P =  \frac{10}{25}=\frac{2}{5} .

As a % =  100*\frac{2}{5}=40 %

D

IanStewart · Answer

It's only correct to calculate probability in this way if each event is equally likely, and as I would read the question, some final locations are more likely than others for the person. When I read the question:

if the person truly starts in a random location between M and U, then there's a 30 mile stretch where he could start, so that must be the denominator of the probability. Now, it's not clear what the person does when they start within 5 miles of M, but since the person travels towards M "without passing the city", I'd just assume they stop there. If that interpretation is correct, then as long as the person starts within 15 miles of M, the person will end up within 10 miles of M, and will thus be within 5 miles of the "25 miles from U/5 miles from M" sign. Under that interpretation, the answer is 15/30 = 1/2.

If the question designer wants the answer to be 2/5, then the question isn't properly worded, because in that case, the person does not "start at a random location on the road", but that's what the question tells us is happening. What is the source?

chetan2u · Answer

I would agree it is not worded properly and that is why ‘without passing the city’ could easily mean that he picks a random location but does not pass the city even after travelling 5 miles towards M. That would mean he does not start between 25 and 30. The answer would again be 2/5 here.

For the answer to be 1/2, the question should have been that ‘he travels 5 miles in direction of M or the distance till M if the distance is less than 5.’

The question could have been better worded either way.

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The distance between Urbania and Metropolis is 30 miles, with numbered

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The distance between Urbania and Metropolis is 30 miles, with numbered

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