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Independent Hunters and Lucky Ducks

Date: 01/24/2002 at 10:07:31
From: Nanette
Subject: Probability

I would like to use this problem as a Problem of the Week.

Ten Montana hunters are in a duck blind when 10 ducks fly over. All 10
hunters pick a duck at random to shoot at, and all 10 hunters fire at the
same time.

Assuming Montana duck hunters area all perfect shots, how many ducks could
be expected to escape, on average, if this experiment were repeated a
large number of times?

I have presented this problem to many high school mathematics teachers,
and we are all stumped. Using many sample spaces, Pascal's Triangle, etc.,
the closest solution I can come up with is 4.99.

I sincerely hope you can help me understand what I'm doing wrong.

Thank-you, Nanette

Date: 01/24/2002 at 19:15:11
From: Doctor Anthony
Subject: Re: Probability

You can model this as a problem of placing ten numbered balls at random
into 10 numbered boxes. A box with at least one ball would correspond to a
dead duck (someone with perfect aim shot at it).

Suppose there are p people and N ducks. From the point of view of a
particular duck, the probability that someone will shoot at that duck is
1/N. We have p independent trials, so this is binomial probability.

The expected number shooting at any duck is p/N.

The probability that no one shoots at that particular duck is

   P(0) = (1 - 1/N)^p
        = [(N - 1)/N]^p

The expected number of ducks with no one shooting at them is

        =  N.[(N - 1)/N]^p

            (N - 1)^p
        =  -----------
            N^(p - 1)

If p = 10 and N = 10, this expression gives

   P(0) =   ------  =  3.48678

So on average, if this experiment were repeated a large number of times,
we could expect 3 and 1/2 ducks to escape.

- Doctor Anthony, The Math Forum   
Associated Topics:
High School Probability

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