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Shooting Ducks Problem

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  • Shooting Ducks Problem

    Could someone explain the following problem to me:

    A hunter hits a duck with a probability of .3 What is the probability that he needs 10 or more shots to get 2 or more ducks.

  • #2
    Originally posted by sateach
    Could someone explain the following problem to me:

    A hunter hits a duck with a probability of .3 What is the probability that he needs 10 or more shots to get 2 or more ducks.
    The basic (but long way to do it) is:

    Total probability = 1 - [P(2 shots)*P(2 or more ducks) + P(3 shots)*P(2 or more ducks) + .... + P(9 shots)*P(2 or more ducks)]

    = 1 - [P(2 shots)*(1 - P(0 or 1 ducks)) + P(3 shots)*(1 - P(0 or 1 ducks)) + .... + P(9 shots)*(1 - P(0 or 1 ducks))]

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    • #3
      Originally posted by wat
      The basic (but long way to do it) is:

      Total probability = 1 - [P(2 shots)*P(2 or more ducks) + P(3 shots)*P(2 or more ducks) + .... + P(9 shots)*P(2 or more ducks)]

      = 1 - [P(2 shots)*(1 - P(0 or 1 ducks)) + P(3 shots)*(1 - P(0 or 1 ducks)) + .... + P(9 shots)*(1 - P(0 or 1 ducks))]
      Thank you, but what would your answer be?

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      • #4
        Originally posted by sateach
        Thank you, but what would your answer be?

        Sorry - I'll have to edit my answer:

        For all the terms in the brackets that deal with number of shots, it shouldn't be "P(n shots) * P(2 or more ducks)". It should be "P(2 or more ducks | n shots)".

        So, revising my answer, it should be:

        = 1 - [(1 - P(0 or 1 ducks | 2 shots)) + (1 - P(0 or 1 ducks | 3 shots)) + .... + (1 - P(0 or 1 ducks | 9 shots))]

        = 1 - [(1 - (1*(.7^2) + 2*(.3*.7))) ....]

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        • #5
          You know what? I take my entire suggestion. I think this problem follows a different distribution. Offhand, I'd guess some form of the hypergeometric distribution (mth success in n tries).

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          • #6
            P(exactly 0 ducks in first 9 shots)+P(exactly 1 duck in first 9 shots)=

            .7^9+9*.3*.7^8= whatever that computes to.

            It is a tricky question because of the "2 or more", but seeing as he shoots (or misses) ducks senquentially he satisfies the "2 or more" at the moment he hits his second duck. Therefore there is no need to waste any time on the "or more" part.

            Goodluck.

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