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# Once every three years is surreal, I'm told...

On December 29th, there were five bank robberies in New York City. The NPR bit on it contained this tidbit:

Anahad O'Connor, metro and breaking-news reporter for The New York Times, says the news was a little surreal because last year there was fewer than one bank robbery a day in the city.

So, how surreal is it, actually? According to O'Connor's article, there were 280 bank robberies in the first 362 days of 2007. So, there is an average of 280/362 bank robberies per day in NYC (at least for most of 2007).

Now, it would be hard to believe that bank robbers don't have preferred days of the week. For example, outside of Hollywood films, it is probably much more likely that a bank gets robbed while there are tellers on duty than on days when the bank is closed. It is technically easier to get a person to give you money than to get a 10 inch steel door to give you money.

So, there are probably some days where the raw likelihood is higher than other days. But, for the sake of argument, let's say that all days are equiprobable since this makes five robberies in one day as improbable as it can be with still an overall average of 280/362 robberies a day.

Similarly, any conspiracy to rob multiple banks would probably tend to cluster them for the same time period so as to thin out the police response. So, let's assume that each robbery is entirely independent of the others.

Under these minimizing assumptions, the number of NYC bank robberies on a given day is a Poisson distributed random variable with λ = 280/362. Thus, the probability of five robberies in a given day is a bit over 1 in 1000. [ λ5e-&lambda/5! ~=~ 0.0010645 ] That's a bit more than once every three years. That's just not surreal.

And, it's way less surreal than all of that. The same New York Times article says that bank robberies were up 54% at that point in 2008. There were 431 bank robberies in the first 362 days of 2008. Using 431 in place of 280 in the above calculations brings the probability up to almost 6 in 1000 (0.0059982). That's twice a year, on average.

The moral: probability and gut instinct are not close allies.

Maybe 2008 having 431 bank robberies in the same time that 2007 had 280 wasn't as improbable as it sounds either. If we assume the real average is 280 bank robberies in the first 362 days of a year, what are the odds of finding a year with 431 bank robberies in the first 362 days?

It turns out, it's less than 1.5 in 1017. This is roughly once every 10 million age of the universe's.

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