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Ten Ducks and Ten Hunters

Date: 02/03/2000 at 09:00:29
From: Amal
Subject: Probability 

Ten ducks fly overhead. Each of ten hunters picks one duck at random 
to aim at and kill it with probability P. What is the mean number of 
ducks that are killed?

Date: 02/03/2000 at 18:08:53
From: Doctor Anthony
Subject: Re: Probability 

If we consider the fate of one duck we have the possibilities of 0, 1, 
2, ..., 10 guns firing at it. The probabilities are binomial with 
n = 10.

The probability of a particular gun aiming at the duck = 1/10

     P(0) = (9/10)^10
     P(1) = C(10,1)(1/10)(9/10)^9
     P(2) = C(10,2)(1/10)^2.(9/10)^8    and so on

With 1 gun the probability of being killed is P(1).p

With 2 guns the probability is  P(2).[2p-p^2]

With 3 guns the probability is  P(3).[3p-3p^2+p^3]

With 4 guns the probability is  P(4).[4p-6p^2+6p^3-p^4]   and so on.

We continue up to the situation where all 10 guns are aimed at one 
unfortunate duck. The sum of these probabilities is the overall 
probability that a duck is killed.

Finally, the expected number of ducks killed will be the sum of all 
these probabilities multiplied by 10.

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

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