# [extropy-chat] calling all bayesians

Eliezer S. Yudkowsky sentience at pobox.com
Thu May 12 03:57:18 UTC 2005

```spike wrote:
> Guys help me eff this real-life effing problem:
>
> I build 150 droobs and use 131 of them in my freem.  I
> test the remaining 19 spares destructively and find that
> all are good.  From that information only, what is the
> probability that all 131 droobs are good?
>
> I have four Monte Carlo sims chewing on this problem
> but they are giving me puzzling results.  A closed-form
> solution to this would be impressive, winning my
> undying respect.

If your prior belief is that any number of good droobs between 0 and 150 is
equiprobable, then this is the *classic* Bayesian problem, the one that Bayes
himself considered.  If I recall correctly, the closed form solution *for this
prior* is that if you observe X good cases and Y bad cases, the posterior
expected probability of goodness is:

X + 1
-----
X+Y+2

Thus if you started out believing that any failure rate between 0 and 150 was
equally plausible, you would expect that the probability is 20/21 that any
given remaining droob is good.

You actually asked a more difficult question, the probability that all
remaining droogs are good.  Intuitively I would expect the answer to be 20/21
* 21/22 * 22/23 ... = 20/152 but I haven't checked my work.

Other priors give different answers.  I don't think the equiprobable prior is
a reasonable one for this case; we don't think it equally likely that a
manufacturing facility turns out 150 straight successes vs. 150 straight
rejects.  In real life, Spike, your problem is pretty much undefined, unless
you can give me some kind of base rate statistics on how often your
manufacturing technique works.

How on Earth did you set up a Monte Carlo sim on this?

--
Eliezer S. Yudkowsky                          http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence

```