I have hit a brick wall in a problem I'm trying to solve. So far I've derived the following equation:
0.05 = (t+1-n)! / (w+1-n)! where t>w
Could someone take pity on slow old brain and teach me how to solve for 'n'?
Ok, the problem I'm trying to solve is more of a probability problem than a number theory one, but the place I got stuck on was to do with factorials so I posted it here. The full problem I'm trying to solve is this::
There is a jar with T balls in it, W of the balls are white, B balls are black so T = W+B. A person pulls out one ball at a time, and does not replace the ball.
What I would like to do is estimate how many trials (n), on average, it would take for a person to find a black ball if they were to repeat this process, say, for 100 jars of balls? That is, if a person was to repeat the process (randomly pull balls out of the jar until a black one is found, record n, then start again with a new jar), what should the average number of attempts be before a black ball is found in each jar?
My approach so far has been to calculate the probability of going for n trials without selecting a black ball.
Probability of selecting white balls for n trials = PWn = W/T * (W-1)/(T-1) * ... * (W-(n-1))/(T-(n-1)) = (W!/(W-n)!) / (T!/(T-n)!)
So if I select a low level of probability (and thus a high level of confidence that a B ball will be selected), can I estimate how many trials (n) will be needed for this level of probability.
i.e., solve PWn = 0.05 = (W!/(W-n)!) / (T!/(T-n)!)
Looking back through my working out, I can see that the equation I had in my original post above is an incorrect attempt to simplify this. And hopefully, someone here can either teach me how to solve for n, or perhaps suggest another approach for my original problem.