Seeing the same truck

My friend, who writes novels, posed the following to me:

"I'm stuck on something for this thriller I'm writing. I have this character who can roughly compute probabilities off the top of his head. He's smart more than autistic, if you know what I mean.

The guy sees a truck (he knows the plates) parked outside his house,  and then sees the same truck show up again in a separate part of the country later. What are the odds of it being a coincidence?

Say there are 150,000,000  private homes in America. And only one of these trucks has the number plate.
Does that make the odds 150,000,000 to 1?

Or this is all completely incalculable?"

The problem is not very clearly stated and my friend’s suggested answer is an example of the same kind of false reasoning explained in detail in this classic example. Here is the reply I gave:

The probability is definitely not 1 in 150 million as you suggest. It is a little complicated to get to a proper answer, but here goes:

What you are saying is that there is definitely only one truck with this number plate. What we want to calculate is the probability of this guy (let's call him Phil) seeing the same truck in two completely different locations within a given period of time. First thing to note is that you have to avoid the common fallacy of overstating the 'coincidence' by ignoring the fact that there are many unique trucks which are candidates for being seen twice in different locations. Suppose, for example, that in any one year  there are 1000 different vehicles parked near this guy's house. What you have to calculate is the probability that ANY one of them might subsequently be seen by the guy in a different location. That is 1000 more times likely than the probability of one SPECIFIC vehicle being seen twice.

So we have to first work out the probability of a specific vehicle V being seen twice by Phil and then multiply that probability by 1000.

The probability of Phil seeing the specific vehicle in two completely different locations depends on

a)    the amount of travelling Phil does and
b)    the average amount of travelling an arbitrary vehicle might do.

If, for example, Phil NEVER visits any other city than his own then the probability of him seeing vehicle V in two cities is actually 0 (which is even less than 1 in 150 million of course!)

But let's suppose on average that Phil visits 10 different cities within the same year (or whatever period you want to restrict it to). Let's suppose that at any time there are 100,000 vehicles in each city and that on any visit let's suppose that Phil sees 1 in every 100 cars in the city (i.e. he sees 1000 vehicles).   Now we have to consider the probability that Phil sees vehicle V if it is in any one of those 10 cities at the same time as him. This is more or less 10 times the probability that Phil sees vehicle V in one specific other city C. To work out this probability we can assume something like the following:

Now, as I said, there are 10 specific cities where they could meet so we need to multiple the last probability by 10.

The result is a probability of 1 in 35,000

Finally there were 1000 candidate vehicles to choose from so we need to multiple the last probability by 1000.

That leaves a 1 in 35 chance of seeing the same vehicle in a different city.

It's still unlikely of course, but  MUCH less unlikely than the 1 in 152,000,000 you came up with.

Obviously lots of assumptions and approximations but the point is that your initial estimate is several orders of magnitude out.

This all helps to explain why so-called one in a million events happen all the time. It's because there are millions of events that can happen!

As I said, it is probably useful to look at a simpler example like the one here to really get your head around this.

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Norman Fenton