If Muslims are 1⁄4 of the world’s population then knowing someone is Muslim cannot possibly increase Pr(person is a terrorist) by more than a factor of 4. (That’s what you’d get if all terrorists without exception were Muslims.)
For your “purple people” analogy to apply in a world where 1⁄4 of people are Muslim, it would be necessary for 1⁄4 of the world’s population to be terrorists. That is … not close to being credible.
(However, in this context I don’t think the fraction of the world’s population that’s Muslim is actually what’s relevant; what’s relevant is the fraction of the US population, or the fraction of people seeking to move to the US from elsewhere. I believe those fractions are quite a lot smaller than 1⁄4, so discovering that someone in one of those categories is Muslim can give you more than 2 bits of information.)
If Muslims are 1⁄4 of the world’s population then knowing someone is Muslim cannot possibly increase Pr(person is a terrorist) by more than a factor of 4. (That’s what you’d get if all terrorists without exception were Muslims.)
For your “purple people” analogy to apply in a world where 1⁄4 of people are Muslim, it would be necessary for 1⁄4 of the world’s population to be terrorists. That is … not close to being credible.
(However, in this context I don’t think the fraction of the world’s population that’s Muslim is actually what’s relevant; what’s relevant is the fraction of the US population, or the fraction of people seeking to move to the US from elsewhere. I believe those fractions are quite a lot smaller than 1⁄4, so discovering that someone in one of those categories is Muslim can give you more than 2 bits of information.)