Re: Darts probability

On Mon, 21 Aug 2006, JeffCameron wrote:
As long as the distribution of the distance to the bull is the same
for each dart, independent, and continuous (to avoid ties), then the
actual pdf has no effect on the answer.

Interestingly, I drew an even stronger (and initially surprising)
conclusion from my experiments.

You assume above that the probability distribution must be symetrical,
and centered at the bulls-eye. Such probability distributions include
a uniform distribution, as well as a bell-curve centered at the
bulls-eye. However, it turns out that the probability is 2/3 no matter
what probability distribution is used.

My initial hypothesis was that 2/3 would be the probability for all the
symetrical distributions cenetered at the bulls-eye, because of the
effect that you pointed out. However, I did not expect 2/3 when using
arbitrary distributions, not centered at the bulls-eye.

Jeff Cameron

I don't think it makes any difference to the analysis whether it's
symmetric or not. In polar coordinates the pdf is given by some function
P(r, theta). A symmetric pdf would not depend on theta at all, but if it
does, since only the distance from the center (r) enters into the problem,
you can marginalize over theta and get a pdf for r alone. So even for an
asymmetric pdf, you can reduce the problem to just being in terms of r.

-Christopher
.

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