so being Muslim literally only provides two bits of information.
I suggest you read up on the concept of information before commenting further.
If 1⁄4 of people are Muslim, the fact that someone is Muslim gives you two bits to help identify them uniquely. True, but irrelevant. We don’t want to identify them uniquely, we want to know if they are a terrorist.
However, if hypothetically all “purple people” are terrorists and all terrorists are “purple people” then the fact that someone is a “purple person”, or is not, tells you 100% of what you want to know in respect of their being a terrorist. Which is one bit of information: terrorist or not.
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.)
We don’t want to identify them uniquely, we want to know if they are a terrorist.
Even if all terrorists were Muslim, knowing that somebody is Muslim would only increase the probability that they are a terrorist by a factor of four. If there also are non-Muslim terrorists, P(terrorist|Muslim)/P(terrorist) is even less than that.
one bit of information: terrorist or not
“Fifty-fifty, either I win [the lottery] or I don’t.”
A hypothetical in which 50% of the population is terrorists doesn’t sound particularly relevant to the real world. And if the fraction of the population which is purple people is less than 50% (let’s call it p) learning that somebody is a purple person doesn’t only give 1 bit of information, it gives -log2(p) bits, which in order for p to be remotely near the fraction of terrorists in the real world would have to be at least a dozen, I guess.
I suggest you read up on the concept of information before commenting further.
If 1⁄4 of people are Muslim, the fact that someone is Muslim gives you two bits to help identify them uniquely. True, but irrelevant. We don’t want to identify them uniquely, we want to know if they are a terrorist.
However, if hypothetically all “purple people” are terrorists and all terrorists are “purple people” then the fact that someone is a “purple person”, or is not, tells you 100% of what you want to know in respect of their being a terrorist. Which is one bit of information: terrorist or not.
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.)
Even if all terrorists were Muslim, knowing that somebody is Muslim would only increase the probability that they are a terrorist by a factor of four. If there also are non-Muslim terrorists, P(terrorist|Muslim)/P(terrorist) is even less than that.
“Fifty-fifty, either I win [the lottery] or I don’t.”
Can you read?
A hypothetical in which 50% of the population is terrorists doesn’t sound particularly relevant to the real world. And if the fraction of the population which is purple people is less than 50% (let’s call it p) learning that somebody is a purple person doesn’t only give 1 bit of information, it gives -log2(p) bits, which in order for p to be remotely near the fraction of terrorists in the real world would have to be at least a dozen, I guess.