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# A boy is late for school

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Posted 25 September 2013 - 05:45 PM

A boy is late for school often.  When approached by his teacher, he explained that it is not his fault.  Then he provided some details.  His father takes him from home to the bus stop every morning.  The bus is supposed to leave at 8:00 am but the departure time is only approximate.  The bus arrives at the stop anytime between 7:58 and 8:02 and immediately departs.  The boy and his father try to arrive at the bus stop at 8:00 however due to variable traffic conditions they arrive anytime between 7:55 and 8:01.  This is why the boy misses the bus so often.

Can you determine how often the boy is late for school?

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### #2 Anza Power

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Posted 25 September 2013 - 07:14 PM

Is the arrival time continuous or discrete (whole minutes)?

Can we assume the distribution is uniform?

Continuous time is the interesting case so let's look at that with uniform distribution:
Spoiler for

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### #3 witzar

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Posted 25 September 2013 - 09:02 PM

Spoiler for without integrals

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### #4 The_King_Of_Games

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Posted 03 October 2013 - 08:16 AM

witzar has it IMO
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### #5 dgreening

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Posted 14 October 2013 - 06:36 AM

Spoiler for slight correction

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### #6 bonanova

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Posted 14 October 2013 - 09:55 AM

Spoiler for slight correction

The OP does not constrain the boy's arrival to be on the minute.
Between 7:58 and 7:59, for example, p decreases linearly from 1 to 0.8; it does not drop abruptly to 0.8 at 7:58.
Similarly for the next two minute-intervals.
The boy does not arrive later than 8:01.

Make a graph of p vs boy's arrival time and find its average value.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
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### #7 dgreening

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Posted 18 October 2013 - 06:58 PM

Spoiler for slight correction

The OP does not constrain the boy's arrival to be on the minute.
Between 7:58 and 7:59, for example, p decreases linearly from 1 to 0.8; it does not drop abruptly to 0.8 at 7:58.
Similarly for the next two minute-intervals.
The boy does not arrive later than 8:01.

Make a graph of p vs boy's arrival time and find its average value.

good point!

In fact the boy does not necessarily arrive on the minute and the bus does not necessarily arrive on the minute either.

I think your approach is better.

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