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From: dms@dciem.UUCP (Dave Sweeney)
Newsgroups: net.math
Subject: Re: New Probability problems: answers
Message-ID: <412@dciem.UUCP>
Date: Tue, 27-Sep-83 11:40:26 EDT
Article-I.D.: dciem.412
Posted: Tue Sep 27 11:40:26 1983
Date-Received: Tue, 27-Sep-83 12:34:49 EDT
References: <516@houxz.UUCP>
Organization: D.C.I.E.M, Toronto, Canada
Lines: 28

Problem no. 6 (how many people will you next meet before you
meet one born on Sunday?) should be answered in the following way:

Let p = Prob(born on Sunday) = 1/7 on the equidistribution hypothesis,
and q = 1 - p = 6/7.  Let n = number of people met including the last
who is born on Sunday.

Then n = 1 with probability p; n = 2 with probability pq (first person is
not born on Sunday, second person is); n = 3 with probability (q**2)p,
and so on.  The expected value of n is the mean of (n * probability of n),
summed over all values of n from 1 to infinity:

	E[n] = sum from 1 to infinity of (n * (q**(n-1)) * p)

	     = (p/q) * sum from 1 to infinity of (n * q**n)

which reduces after a little algebra to

	E[n] = (p/q) * (q/p**2) = 1/p = 7.

The interested reader can verify with little trouble that the
probabilities themselves sum to 1.

Thus one can expect on the average to meet 6 people not born on
Sunday before meeting the 7th who is.

			Dave Sweeney
			..!utzoo!dciem!dms