# S2 Poisson - any clever ways to work backwards

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#1
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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5 years ago
#2
(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 .

This can be solved using logs.
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#3
(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 .

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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