My ultimate goal is to answer this question: what are the odds that three state licensing boards each made a mistake by licensing an applicant as a psychologist, when they actually should not have done so because the psychologist did not meet an education and training requirement?
But I will start with the same question concerning two licensing boards so that I don't get myself confused. :O/
$P(A)$ = Probability that Maryland mistakenly licenses a psychologist. The mistake is due to an error checking credentials.
$P(B)$ = Probability that Virginia mistakenly licenses the same psychologist. Virginia's mistake is also due to an error checking credentials.
$A$ and $B$ are independent events.
However, if Maryland makes a mistake, we assume for this hypothetical that the error is due to complexity in the applicant's education and training documentation.
Consequently, if Maryland makes a mistake and licenses a psychologist they shouldn't have licensed, then Virginia is more likely to make an error too because of the complexity in the applicant's education and training documentation.
For this hypothetical, we will assume that:
$P(B|A) = 0.2$.
The probability of Virginia making an error, given that Maryland made an error, is $0.2$, not because the Maryland decision influences the Virginia decision (it doesn't), but because we know (in this hypothetical scenario) that the applicant's education and training documentation is more complex than usual, and therefore more prone to error.
Given these hypothetical facts: What is the probability that both Maryland and Virginia will make mistakes and license a psychologist who should not have been licensed?
I think the answer is: $P(A ∩ B) = P(A) P(B|A) = (0.05)(0.20) = 0.01$.
Is there a flaw in my reasoning? In my calculation? What else is important for me to know?
TIA,
Mark
P.S. I searched math.stackexchange.com for various terms and found two somewhat related questions with answers, which I read:
Why P(B|A) is not the same as P(A∩B) if both A & B are independent events?
If A and B are independent events but NOT mutually exclusive, find P(A and B)?
There are other questions with answers that are probably related, but they all use some sort of programming or spreadsheet notation that I don't understand.