How long will you wait hitchhiking on the road?

The hitchhiker’s paradox in a consequence of the Poisson’s law, which describes how the occurences of an event during a period of time are happening, based on the observation of the previous actual occurences in the past.

Quick description of Poisson Law

The Poisson law is used to describe the how a large number of random discrete events is occuring. Based on the observation that these events have occured λ time during a Δt period of time in the past, the poisson law describes the probability that the same event happens only x times exactly during the same period of time in the future.

poisson

This is represented as follows:

lambda

The curve is almost symmetric if lambda is big enough, meaning that if we have observed the event enough times in the past to describe it using the Poisson law.

The hitchhiker’s paradox

The paradox is as follows:

Cars are passing by on a road with a mean interval time of 10 minutes (according to a Poisson process). A hitchhikers arrives to the roadside at a random instant of time and is wondering “How long will I have to wait until the next car shows up?

One would probably think that since the average time between cars is 10 minutes, the waiting duration should be around 5 minutes, which is half the time: sometimes the hitchikers will arrive just after the last car passed, waiting close to 10 minutes, sometimes he would arrive just before the next, waiting less than 1 minutes, and the average waiting time should be 5 minutes.

This works if you are waiting for a scheduled bus which arrives every 10 minutes and is never late, in which case you cannot wait more than 10 minutes when you arrive at the bus stop:

scheduled-bus

Scheduled bus arrival forecast

But this does not work when you are waiting for a random event which you describe with an average based on a past observation using Poisson law:

poisson-cars

Cars on a road following Poisson’s process arrival (@10 minutes)

In this case, if you take an interval of observation that is big enough, you will observe as many arrivals below 10 minutes and as many arrivals above 10 minutes. And you will see that the average waiting time is 10 minutes.

This is because the hitchhiker has more chances to arrive during a long interval time than during a short interval time, as you can see on the diagram above.

 

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