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i need a bus to run so it gives me a 1/2 chance of getting to point B faster than walking. The distance from point A to B is 1.5 miles. the speed of the bus is 25mph and the walking speed is 4mph

Distance = speed x time
Speed by walking 4 miles per hour
Speed by bus 25 miles per hour
Total distance= 1.5 miles
Time it takes to go by walking (distance divided by speed) or
2 miles / 4 mph = 1/2 an hour or 30 mins.
Time it takes to go by bus 2 miles / 25 miles per hour or 1/12.5th of an hour or 60/12.5= 4.8 mins.

The answer is 1.59busperhour but I am not sure how to get to this answer.....any help is appreciated!

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  • $\begingroup$ You seem to have two distances involved, $1.5$ miles and $2$ miles. Which is it? $\endgroup$
    – André Nicolas
    Aug 31, 2013 at 17:29
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    $\begingroup$ 1.59 busperhour is a very strange unit $\endgroup$
    – Dominic Michaelis
    Aug 31, 2013 at 17:45
  • $\begingroup$ its 2 miles sorry and its 1.59 buses per hour $\endgroup$
    – Lindsay Baird
    Aug 31, 2013 at 19:42
  • $\begingroup$ using 1.5mph the time it takes to walk is .375 hours and the time it takes to go on the bus with no waiting is .06 hours. if you dont wait for the bus you are saving .3150 hours the difference between the two. but when i use that in your formula i dont get the solution (1.59)...can you please help me? $\endgroup$
    – Lindsay Baird
    Aug 31, 2013 at 19:50

1 Answer 1

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Assume that the interval between buses is a constant $h$ hours.

If we arrive at the bus stop at a "random" time, the probability that our wait will be $\le \frac{h}{2}$ is $\frac{1}{2}$.

So if $\frac{h}{2}\lt 0.3150$, then with probability $\gt \frac{1}{2}$, we will save time by taking the bus. If $\frac{h}{2}\gt 0.3150$, then with probability $\gt \frac{1}{2}$, using the bus takes longer. The "breakeven" point is $\frac{h}{2}=0.315$. That gives $h=0.63$.

It follows that there are $\dfrac{1}{0.63}\approx 1.5873$ buses per hour if we have probability exactly $\frac{1}{2}$ of saving time by taking the bus.

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  • $\begingroup$ using 1.5mph the time it takes to walk is .375 hours and the time it takes to go on the bus with no waiting is .06 hours. if you dont wait for the bus you are saving .3150 hours the difference between the two. but when i use that in your formula i dont get the solution (1.59)...can you please help me? $\endgroup$
    – Lindsay Baird
    Aug 31, 2013 at 19:51
  • $\begingroup$ @LindsayBaird: My previous answer assumed you were in the exponential distribution/Poisson distribution part of the course. The (greatly modified) current answer assumes a different model, and gives the answer you are looking for. $\endgroup$
    – André Nicolas
    Sep 1, 2013 at 17:16
  • $\begingroup$ im not really getting it...you just doubled the time difference...? $\endgroup$
    – Lindsay Baird
    Sep 1, 2013 at 17:17
  • $\begingroup$ if i multiply .3150 * 1/2 i get .1575 which is close but not quite exact answer in the solution manual which is 1.5873... $\endgroup$
    – Lindsay Baird
    Sep 1, 2013 at 17:19
  • $\begingroup$ @ andre im dumb thanks its been driving me crazy all weekend $\endgroup$
    – Lindsay Baird
    Sep 1, 2013 at 17:21

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