Ten Ducks and Ten HuntersDate: 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 http://mathforum.org/dr.math/ |
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