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From: rjnoe@riccb.UUCP (Roger J. Noe)
Newsgroups: net.math
Subject: Re: Pistachio Probabilities
Message-ID: <499@riccb.UUCP>
Date: Fri, 23-Aug-85 09:58:46 EDT
Article-I.D.: riccb.499
Posted: Fri Aug 23 09:58:46 1985
Date-Received: Sun, 25-Aug-85 04:48:20 EDT
References: <285@ihnet.UUCP>
Distribution: net
Organization: Rockwell International - Downers Grove, IL
Lines: 41

> Suppose you begin with a bag containing O openable pistachios,
> and U unopenable pistachios.  A trial consists of selecting a nut at random,
> and eating it (if possible), or returning it to the bag.
> How many trials, on the average, are required to consume all the "openable"
> pistachios?  Express the answer in terms of U and O.
> Any ideas on the variance/distribution of trials(U,O)?
> Although I have not given this problem a lot of thought (yet),
> it looks surprisingly difficult.
> The pistachios, by the way, were delicious.
> -- 
> 	This .signature file intentionally left blank.
> 		Karl Dahlke    ihnp4!ihnet!eklhad

Actually, it's not that hard.  Express the problem in terms of a recurrence
relation.  Let T(No,Nu) denote the expected number of trials given that there
are "No" openable nuts and "Nu" unopenable nuts in the bag.  Assume also that
we know No before we begin so that we don't keep searching for openable nuts
when there are none left.  Then for No=0, T(0,Nu)=0.  Now, what if No>0?
Clearly there is a No/(No+Nu) probability of selecting an openable nut, which
adds a trial and decrements No.  There is a Nu/(No+Nu) probability of picking
an unopenable nut, which adds a trial but does not change No nor Nu.  (Try
pronouncing "No" as "no" and "Nu" as "new" - these sentences can be a lot of
fun!)  Then the relation for No>0 is

	                No                   Nu
	T(No,Nu) = 1 + ----- * T(No-1,Nu) + ----- * T(No,Nu)
	               No+Nu                No+Nu

Rearranging,

		T(No,Nu) = 1 + Nu/No + T(No-1,Nu)

So the obvious solution is
		                     _No
		                     \
		T(No,Nu) = No + Nu *  >  1/k		(No > 0)
		                     /__
		                     k=1
--
Roger Noe			ihnp4!ihopa!riccb!rjnoe
Rockwell International