## Thursday, July 22, 2010

### I Was Late for That First Meeting to Schedule the Meeting About the Conference

Yesterday morning I ended up waiting for 30 minutes for a bus.  According to the schedule, a bus should arrive at my stop every 10 minutes.  That was the start of a particularly bad day for me.

Today, I woke up and felt like today was going to be better.  I expect when I show up at the bus stop, it will be either right there or come within the first minute.

Am I doomed to disappointment?

1. I love this problem, so many assumptions to make and so many ways to answer!

I'll give one possible answer - assuming the timetable is correct and the buses arrive on time, then at any given minute a bus is either at the stop, or 9,8,7,6,5,4,3,2 or 1 minutes away from the stop - a total of ten different possibilities. So a bus being at the stop or one minute away is 2 out of ten possibilities, a 20% chance of meeting your expectation, or an 80% chance of disappointment.

Thanks for all these great puzzles!

2. reallyfatbloke, you're welcome.

The arrival of the bus can be defined as a poisson process, while the time to arrival is on an exponential probability curve. In this case, the average (or expected time) till a bus arrives is 10 minutes. So the probability is about 9% that you will only have to wait for one minute or less for the bus to arrive.

The chances of waiting for at least 30 minutes is about 0.5%.

There's a good explanation of the two concepts here: http://cs.wellesley.edu/~cs199/lectures/19-exponential.html

I know, not exactly a brain teaser, but I find it interesting.

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