MHB False Positive Rate of 1:1.5M Sampling Process

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The discussion centers on the false positive rate of a sampling process where items are classified as type A or B, with a false positive rate of 1 in 1.5 million. When a sample of 1 million yields one classification as type A, the probability of that classification being a false positive is approximately 0.000067. The sample size does not affect this probability, as it remains constant regardless of the number of observations. The conversation highlights the relevance of Bayes' Theorem for more complex scenarios, although it is not necessary in this case due to the known false positive rate. Overall, the key takeaway is that the probability of a false positive remains at 1 in 1.5 million for each individual 'hit'.
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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?
 
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philpq said:
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.
 
I like Serena said:
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 :)
 
Usually questions about false positives use Bayes' Theorem and for that you need a lot more information.

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

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.
 
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