The Math of Waiting

Too long waiting for buses and other uncertain events leads me down this road:

Given an event occurring on average at 1 per hour (m = 1) we can see the probability of the event having occurred after m hours with the exponential distribution.

Some values of p for t:

x	p
0	0
1	0.632121
2	0.864665
3	0.950213

If expecting an event on average every unit of time U then after 3 units of time the probability that your expected event should have happened is greater than 95%. On a 5% confidence limit you can reject your hypothesis that events are following your expected model.

e.g. So if tubes are expected every 5mins on average (and assuming they arrive randomly!) then after 15mins we can reject this idea that they are arriving every 5mins… something has most probably happened to disrupt the service.

If you take tubes on this random line regularly then 63% of the time you would have boarded a tube after 5mins, and 86% of the time after 10mins.

Conversely: if you have been waiting for less than 0.0512933 units of time and an event happens then you might want to review ideas about the event happening every unit of time.

And if an event from an unknown distribution happens then you know the average unit of time is at least 1/0.0512933 = 19.5 times as long as you just waited.

In the case of the recent Russian plane crash over Sinai. It has happened a little over 1 month after starting operations in Syria. If the two events are lined then Russian plane crashes must happen no more than every 19.5 months.

In reality this model is overly simple as such events vary around a mean of the expected time. But as a crude rule of thumb 3 x the unit of time is a start.