[extropy-chat] calling all bayesians

[Dan Clemmensen]

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.
>
>spike
>
>
>
>  
>
As stated, a droob must only be used once and will either work or not,
like a hand grenade, not like a hard disk.

We start by computing a probability of failure and a confidence interval
for the test of the 19 droobs.
   Probability of success =p.
Example: if p =.5, then the chance of 19 of 19 successes is one in 
2^^19, or one in about 500,000.
better stated,  chance of 19 successes is .5^^19.

If p=.99  (i.e. 99% reliable) the chance of 19 successes is .99^^19.

Let's use 99% reliability as our guess. Then the chance that one of the 
130 will fail is:
  .99^^130, Which is .27

All of the above is elementary probability theory. not statistics, and I 
no longer remember
the statistical theory to combine the confidence interval back with this 
last number, so let's
try this: let's assume that the best estimator for p is the one with a 
50% confidence:
     p^^19=.5
   p= 19th root of .5, or about .965. That is about half the time we 
test a set of 19 droobs, all of them
will pass if there is a 3.5% failure rate.

Now, .965^^130=.0097 or thereabouts.