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Math Help - Tree Problem

  1. #1
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    Tree Problem

    The Spruce tree drops seeds every fall with its leaves, 80% of the seeds survive the winter. For a seed that survives its first winter there is a 40% chance it sprouts and a 40% chance it remains dormant and a 20% chance it dies each winter. About 60% of the sprouts survive to become mature trees capable of producing their own seeds. This sprout to mature tree process takes 4 years. What is the critical number of seeds dropped per tree to ensure the survival of the spruce tree species?
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  2. #2
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    Quote Originally Posted by mathlete2 View Post
    The Spruce tree drops seeds every fall with its leaves, 80% of the seeds survive the winter. For a seed that survives its first winter there is a 40% chance it sprouts and a 40% chance it remains dormant and a 20% chance it dies each winter. About 60% of the sprouts survive to become mature trees capable of producing their own seeds. This sprout to mature tree process takes 4 years. What is the critical number of seeds dropped per tree to ensure the survival of the spruce tree species?
    The probability that a single dropped seed becomes a mature tree is (0.8)(0.4)(0.6) = 0.192.

    Let n seeds drop and let X be the random variable number of seeds that become a mature tree.

    Then X ~ Binomial(n, p = 0.192).

    It now depends on how you define in, a probability sense, the statement ensure the survival of the spruce tree species .......

    Some might argue that you now require E(X) = 1 ..... Others might argue that you require \Pr(X \geq 1) > 0.99, say ........

    You'll have to decide for yourself (or get clarification from your instructor).
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