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    I had a question where I had to work backwards and found an approximate answer but it was off by quite a bit. So I was wondering if there is a more efficient way to find the value I need.

    The question is:

    1 person in 80 has blood type P.

    In first part of question a sample of 150 people is taken, hence mean is 150 * 1/80 = 1.875 and so you can use X~Poisson(1.875) to calculate probabilities.

    But the last part of the question asks:

    A hospital urgently requires blood type P. How large a random sample of donors must be taken in order that the probability of finding at least one donor of type P should be 0.99 or more?

    This is how I attempted it.

    We need P(X >= 1) >= 0.99

    and Poisson(x/80, 0) <= 0.01

    I just did it trial and error entering mean as 1,2,3,4,5 and got to a probability that matched.

    I got to Poisson(5, 0) = 0.00674 for answered that 5 * 80 = 400+ should be sampled.

    But this was not very accurate and optimal value is 369.

    ie Poisson(369/80, 0) is perfect value.

    Is there a more efficient way to find the value than just trying lots of values???
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    (Original post by acomber)
    I had a question where I had to work backwards and found an approximate answer but it was off by quite a bit. So I was wondering if there is a more efficient way to find the value I need.

    The question is:

    1 person in 80 has blood type P.

    In first part of question a sample of 150 people is taken, hence mean is 150 * 1/80 = 1.875 and so you can use X~Poisson(1.875) to calculate probabilities.

    But the last part of the question asks:

    A hospital urgently requires blood type P. How large a random sample of donors must be taken in order that the probability of finding at least one donor of type P should be 0.99 or more?

    This is how I attempted it.

    We need P(X >= 1) >= 0.99

    and Poisson(x/80, 0) <= 0.01

    I just did it trial and error entering mean as 1,2,3,4,5 and got to a probability that matched.

    I got to Poisson(5, 0) = 0.00674 for answered that 5 * 80 = 400+ should be sampled.

    But this was not very accurate and optimal value is 369.

    ie Poisson(369/80, 0) is perfect value.

    Is there a more efficient way to find the value than just trying lots of values???
    You can solve this using the binomial distribution. You need the probability that no type P donor is found to be less that 0.01.

    This is \left( \dfrac{79}{80} \right)^n&lt;0.01.

    This can be solved using logs.
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    (Original post by BuryMathsTutor)
    You can solve this using the binomial distribution. You need the probability that no type P donor is found to be less that 0.01.

    This is \left( \dfrac{79}{80} \right)^n&lt;0.01.

    This can be solve using logs.
    Where does the 79 come from?

    OK, I think I have got it. Because r = 0 you can do as:

    nC0 (79/80)^n-0 (1/80)^0

    nC0 = 1

    so simplifies to 79/80^n as you stated.
 
 
 
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