(urth) Silk vs. Cthulhu

[Gwern Branwen]

On Sat, Jun 5, 2010 at 11:16 AM, Stuart Hamm <hammstu at sbcglobal.net> wrote:
> So today I found my car with it's "Silk for Calde" bumper sticker parked behind a car with a sticker that read "Got Cthulhu?"
> What are the odds??

Ooh! This sounds like a fun Fermi problem.*

We can use the Birthday Paradox to find out. One result in the
Wikipedia article is that

'This is a result of the good approximation that an event with 1 in k
probability will have a 50% chance of occurring at least once if it is
repeated k ln 2 times.'**

Let's assume you park once a day averaged over your lifetime and you
start driving at age 20; your life expectancy would be somewhere
around >70, so that's 50 years * 365 days per year, so that's 18250
days, and one parking per day so 18250 parking incidents where you
could've noticed such a thing.

Let's assume that there was a 50% chance of you ever seeing such an
occurrence (you were somewhat lucky), so that we can say '18250 = k *
ln 2'. Then, '18250 / ln 2 = k', and a calculator says that k = 26329
and so the odds are 1 in 26,329.

Of course, this is only the odds of *you* seeing someone with a
Cthulhu sticker parked in front of you. You observe a Silk is Calde
sticker every day, after all. For the rest of us, the odds are even
worse since our observation would have to both stickers. So the odds
for us is probably more like 1/26,329 * 1/26,329. (For you, it's more
like 1 * 1/26,329 as explained previously.)

In an aside, there are 254 million cars in the US.
(http://en.wikipedia.org/wiki/Passenger_vehicles_in_the_United_States)

* http://en.wikipedia.org/wiki/Fermi_problem
** http://en.wikipedia.org/wiki/Birthday_paradox#Approximation_of_number_of_people

--
gwern