# On the probability of running into people

Mrs. G. and I were up in San Francisco last weekend and while on our way to Fog City News we ran into someone we knew. This was sort of surprising, so I got to thinking about how probable it was (or wasn't). Grossly oversimplifying, my reasoning goes something like this:

The population of San Francisco is about 800,000. Let's call it 10^6. I know perhaps 100 people in the city at any given time. There are maybe 20-50 people on any given stretch of city block. Say I walk for an hour at 3 mph and that the average block is 100m long, so I walk for 50 blocks in that time and pass on the order of 10^{3} people. If we assume people are randomly distributed (this is probably pessimistic, since I know that I spend most of my time in SF in a few places and I assume my friends tend to be somewhat similar) then I have a .9999 chance of not knowing any given person I run into. If we assume that these are independent events then I have a .9999^{1000} chance of not knowing any of those people [technical note: this is really (999900/1000000) * (9998999/999999) * ..., but these numbers are large enough and we've made enough other approximations that we can ignore this.] .9999^1000 = .90 so if I walk around the city for an hour, I have about a 1/10 chance of meeting someone I know. That doesn't sound too far out of line.

One assumption you seem to be making is that you are in fact observing every person that you "encounter" as you walk, and in sufficient detail to recognize them. Or that you are similarly observed. Not sure how this would affect your math, nor if it's necessarily quantifiable. I sometimes walk while in "head thought" and not really looking at people as I walk. Other times, I'm into looking at everyone as part of the walk.

I think the selection bias is probably a VERY strong effect, but I think 20-50 is a heavy over-estimate for the number of pedestrians you pass on a given city block (unless you're only in the shopping areas downtown or something)

### October 2012

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