# What sample size to use to reach a 95% or 99% confidence level with poisson

Let say I do an experiment were I get a pass or fail result. I want to know how much single trials (sample size) I should do to get a certain confidence level?

So if the answer that I'm looking for is for example 300 units I got to test, I will test these 300 units and be sure at a confidence level of 95% that my process won't have a failure again.

I started with the poisson law, which could give me the probability that less than 1 fail will happen in the future, but how does the sample size effects the results?

In the scenario you describe, rather than Poisson I would read this as $N=300$ independent observations of a binary outcome [pass/fail]. This would mean that your sampling process has a $Y \sim$ Binomial$(N, p)$ distribution, where $p$ is the probability of a failure.

Now,

$P(Y \lt 1 | N, p) = (1-p)^N = 0.95$

$N = \ln(0.95)/ \ln (1-p)$

While this is the relationship between sample size and the chance of getting a failure it's not actually very helpful. Because it's easy to imagine that whatever the failure rate, if we take a big enough sample we're almost sure to get a failure. I.e. $P(Y \lt 1 | N, p)$ goes to zero with $N$.

Instead, I suppose you have an idea or an agreement what $p$ should be at most; e.g. say $p=0.01$. If that's the case you could calculate the largest acceptable number of of failures by

qbinom(0.95, 300, 0.01) #6


But this doesn't tell you what sample size to use.

If you want to do a sample size calculation you'd need to additionally specify how accurate you need the estimated proportion $p$ to be. Then you could use the formulae in [1].