False Positive Rate of 1:1.5M Sampling Process

[karamand]

I have a sampling process of a very large population in which all items are of type A or type B. I have an analysis of the sampled objects which classifies type A and gives the wrong identification (a false positive) 1 in 1.5 million times.
I take a sample of 1 million and find 1 'hit' i.e classified as type A. What is the probability that it is a false positive?

 

[I like Serena]

I have a sampling process of a very large population in which all items are of type A or type B. I have an analysis of the sampled objects which classifies type A and gives the wrong identification (a false positive) 1 in 1.5 million times.
I take a sample of 1 million and find 1 'hit' i.e classified as type A. What is the probability that it is a false positive?

Hi philpq! Welcome to MHB! :)

Without more information, any 'hit' of type A has a probability of $\frac{1}{1.5\cdot 10^6} \approx 6.7 \cdot 10^{-5}$ of being a false positive.
We will still know basically nothing about the other 999999 observations without more information.

 

[karamand]

Hi philpq! Welcome to MHB! :)

Without more information, any 'hit' of type A has a probability of $\frac{1}{1.5\cdot 10^6} \approx 6.7 \cdot 10^{-5}$ of being a false positive.
We will still know basically nothing about the other 999999 observations without more information.

Thanks for your help. I suppose the answer is obvious when I think about it. The sample size is irrelevant. The probability of anyone 'hit' being a false positive is 1 in 1.5 million as stated :)

 

[Jameson]

Usually questions about false positives use Bayes' Theorem and for that you need a lot more information.

\(\displaystyle P(+|\text{ (actually negative)})=\frac{P(\text{(actually negative)}|+) \cdot P(+)}{P(\text{actually negative})}\)

In the above, $+$ means "reads positive". However, you already have this probability so the above isn't necessary to calculate. I'm just pointing out that these topics are very often related. Here is an example "false positive" question you can read on Wikipedia.

 

[CellCoree]

The probability of a false positive in this scenario would be 1 in 1.5 million, which is the same as 0.000000067%. This is a very low probability, indicating that the sampling process has a high accuracy in identifying type A objects. However, it is important to note that the accuracy of the sampling process may vary depending on the size of the population and the proportion of type A and type B objects. It is also important to consider other factors that may affect the accuracy of the analysis, such as the methodology used and potential sources of error. Further studies and validation of the sampling process may be necessary to ensure the reliability of the results.