Question

It is known that a bus will arrive at random at a certain location sometime between 3:00 P.M. and 3:30 P.M. A man decides that he will go at random to this location between these two times and will wait at most 5 minutes for the bus. If he misses it, he will take the subway. What is the probability that he will take the subway?

Answer 1

The bus and man can arrive in any time between 3:00 and 3:30.

Probability of man arriving before 3:25 = 5/6 as each 5 minute interval is equally possible for his arrival.

Probability of man arriving before 3:25 and catching the bus = 5/6 * 1/6 = 5/36 (as the probability for bus to come in any 5 minute interval is 1/6).

Probability of man to come after 3:25 = 1/6

Probability of man coming after 3:25 and catching the bus = 1/6 * 1/6 *x = x/36, where x is the probability that man comes first. Since both man and bus are equally likely to come first, x = 1/2, giving required probability = 1/72.

So, Probability that man catches the bus = 5/36 + 1/72 = 11/72.

Probability that he takes subway = 1 - 11/72 = 61/72 = 0.8472.

Comments

@Arjun sir in the question we clearly have two random variables, one is arrival time of bus and another is arrival time of man.

But here we say that the man will wait for the bus for at most 5 mins. 

Isn't there an ambiguity as if we say this, how do we know that the man waiting for the bus is independent of the arrival of the bus or is dependent on the arrival of the bus. 

What i mean is that even without the arrival of the bus lets say he would wait for 2 mins and leave -> this is also possible right. 

Then the man's wait time would be another independent variable with range 0 - 5 mins. 

Is this ambiguity a reasonable doubt from my side or can we rule this ambiguity out in the question itself when reading? 

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Bhai top krne ke baad mujhe yaad rkhna🙏

 

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How can you divide 30 minutes into 6 intervals of 5 minites each.

What if he arrives at 3:02 or if he arrives at 3:29 and waits till 3:34.

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I think according to que he will not gonna wait past 3:30

 

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@Prem_Kaushik @Akshdeep I thought about it in a very same manner. This question is a bit poorly framed because certain nuanced assumptions have been implicitly accepted by it. 

To better understand you can think that we are finding 1 - Prob(Man catches the bus). Now certainly if he wants to catch the bus he will wait -> until bus arrives or 5 mins pass, whichever happens first. Now read my answer.

Answer 2

To better understand you can think that we are finding 1 - Prob(Man catches the bus). Now certainly if he wants to catch the bus he will wait -> until bus arrives or 5 mins pass, whichever happens first. 

So two cases exist either that the waiting time lies inside in the period 3:00 to 3:30 or partially outside this period.

example for case 1: man arrives at 3:10 so bus has 5 minutes from 3:10 to 3:15 to arrive. probability of catch = 5/30

example for case 2: man arrives at 3:28 so bus only has 2 minutes to arrive. probability of catch = 2/30

Note that probability of catch at 3:30 is 0 because the points are from 0 to 30 means when 30 started not completed. so basically from start of 0th min. to end of 29th min we have probability of catch. 

Now let man's arrival time be X (or x).

So P(catch|X) =  =    

Therefore : 

AI was used in calculation part to keep it crisp (approach,theory,explanation and thought process are exclusively mine).