# Calculating the probability of a rare event

I've read in the news that the last month five pedestrians died from a population of 500,000 Remembering that it is a poisson problem (the famous prussian horse kicks by Ladislaus Bortewicz). I fired up R to understand the probability

ppois(5*12, 3, lower.tail=F)


since in a month died 5, I multiplied it by 12 months and compared it to the official statistics of 3 pedestrians a year in this area. The result:

[1] 1.310782e-56


So, yes this is an unusual case. Am I doing everything right?

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Let's do a back-of-the-napkin reality check. Three deaths per year is a mean rate of 1/4 per month, so--assuming independence--five deaths in a given month has a chance of (1/4)^5 = 1/1024. I would therefore expect there to be some month in which five deaths occur in any record comprising 1024/2 = 512 months or more, which is about a 40 year period for a single city (population). The news organization probably monitors statistics from dozens if not hundreds of cities; if so, this event is unremarkable. None of these calculations support a number like $10^{-56}$: that's a mistake. – whuber Jan 14 '13 at 15:12
The newspaper is local, so the event is interesting for them, they had the photos of the five victims (one is a four years old girl) on the front page. – Roland Kofler Jan 14 '13 at 15:28
OK, that perspective helps. But consider, too, that your calculations assume pedestrian deaths are a homogeneous Poisson process. They surely are not: deaths will occur in clusters related to weather, to groups of people affected at once by an accident, and so on. Your calculations, even when correctly done, merely support this contention of inhomogeneity. If you are interested in whether these five deaths should be considered unusual, you will need to consult the actual record of pedestrian deaths in this city. – whuber Jan 14 '13 at 15:33
That "back-of-the-napkin" calculation isn't an accurate heuristic. In a Poisson distribution with small mean it overestimates the probability by about a factor of $n! = 120$ for $n = 5$. Of course, I don't think a Poisson distribution is exactly appropriate because there are many accidents which kill multiple pedestrians at the same time, such as a car out of control near a crowd. If the $5$ died from different accidents then the factor of $120$ adjustment should be more reasonable. – Douglas Zare Jan 14 '13 at 16:21
yes they died in different accidents – Roland Kofler Jan 14 '13 at 17:38

Well, 60 pedestrians in a year would be a lot, that's right. But you only have a sample of one month, so it would be more appropriate to scale the yearly average number of deaths down, not the observation up:

ppois(5,3/12,lower.tail=FALSE)
[1] 2.738136e-07


Still unusual, but far less so than 60 over an entire year.

And don't forget selection bias. Newspapers by definition report unusual things, because usual things are not news. So if you read something in the newspaper, it would be very unusual for it to be usual.

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I understand that the justification for the scaling is that I would increase any error that is in the sample, hence scaling down leads to lower errors. Right? – Roland Kofler Jan 14 '13 at 15:07
I love your definition of selection bias. – Roland Kofler Jan 14 '13 at 15:08
+1 Scaling the time up is tantamount to assuming the rate of 5 deaths per month is going to continue throughout the year and actually produce 60 deaths. That's simply making up 11/12 of one's data, which is why such a ridiculous answer ($10^{-56}$) could be derived. – whuber Jan 14 '13 at 15:17
but you would argue also that 2.7e-07 is way too high? – Roland Kofler Jan 14 '13 at 15:25
No, I would argue that 2.7e-07 is still way too low. E.g., it's conceivable your city has experienced a cluster of 5 pedestrian deaths every two years: this is consistent with a mean of 3 per year. It would give a value of 1/24 = 4e-1. I doubt the city has a good record going back more than 2000 months or so. This indicates that any monthly "probability" of less than 1/2000 = 5e-4 would have to be based on a probability model derived from a detailed analysis of the historical data. – whuber Jan 14 '13 at 15:37