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Math Help - Probability of biochemical experiment, distribution

  1. #1
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    Probability of biochemical experiment, distribution

    In a biochemical experiment, n organisms are placed in a nutrient medium, and the number of organisms X which survive for a given period is recorded. The probability distribution of X is assumed to be given by

    P(X=k)=\frac{2(k+1)}{(n+1)(n+2)}\ for\ 0 \leq k \leq n,

    and 0 otherwise

    1) Calculate the probability that at most a proportion \alpha of the organisms survive, and deduce that for large n this is approximately \alpha^2

    2) Find the smallest value of n for which the probability of there being at least one survivor among the n organisms is at least 0.95
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by acevipa View Post
    In a biochemical experiment, n organisms are placed in a nutrient medium, and the number of organisms X which survive for a given period is recorded. The probability distribution of X is assumed to be given by

    P(X=k)=\frac{2(k+1)}{(n+1)(n+2)}\ for\ 0 \leq k \leq n,

    and 0 otherwise

    1) Calculate the probability that at most a proportion \alpha of the organisms survive, and deduce that for large n this is approximately \alpha^2

    2) Find the smallest value of n for which the probability of there being at least one survivor among the n organisms is at least 0.95
    1) Probability that at most a proportion \alpha survives is the same as the probability than at most the number that survive is \lfloor \alpha n \rfloor survive

    This is:

    p=\sum_{k=0}^{\lfloor \alpha n \rfloor} P(X=k)

    and the rest is just algebra.

    CB
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