You blindly draw a ball from the bag; do you know anything about the likelihood of that ball having property X? You now inspect it and find it’s green. Do you now know anything different about that likelihood? If I offered you “pay $100 to win $900 if you drew a property X ball, with the option to double your wager after seeing the color”, you would be happy indeed to see that you’d drawn a green ball and would double your wager.
Regardless of the colors, ratios, and desirability of property X, it seems to me like you do know something new about the likelihood after learning the color.
It also seems to hold if there are 300 million balls, 10 balls with property X, and 3 of those are green, just at a strong cause to believe “this specific ball does not have property X” overall (such as the case in the people examples you gave).
- Nearly all murderers are men.
- Bob is a man.
- Therefore Bob is probably a murderer.
You are mixing logic and probability.
"its not really useful for making judgements about individuals." is not equal to "tells you nothing"
Sticking with medical examples, the first matters for "does this guy have this disease?" The second matters for "how many doses should we order?"
Or, more controversially, "how many of the doctors we train today will still be working in 30 years?" Which came up as a question regarding the increasing proportion of women in medical school. And many people were loudly offended that this might be something to take into consideration. Because they read it as the first kind of question, "will this individual...". But that's different.
Sometimes testing positive on a fairly accurate test for a very rare condition means you are still more likely to not have the condition, and even Doctors sometimes screw that logic up.
Knowing that this very rare disease affects 1 in 10_000 men, and 1 in 100_000 women, doesn't tell you that much about how to perform diagnosis on any one individual. But it does tell you accurately that the male ward probably needs 10x as many beds as the female ward.
Statistics are hard. At least for me.
Edit: also I wasn’t trying to respond directly/refute your comment. Just chiming in with a related concept.
Then we know what mix of coffins to load on the UN plane. But as a doctor on the ground, after your 99% accurate test comes back, you certainly need more information, and knowing the sex of the patient is not much help. The doctor, and the guy loading the plane, are asking very different questions.
Only if you're given the marginal probability ahead of time ;-). Marginals are usually intractable to compute, and people tend to vastly underestimate them. Priors have a similar issue: base rates for a lot of "interesting" conclusions are often really, really low (which is, in a way, why we'd be interested in estimating the posterior!).
Let's estimate the most controversial thing possible: P(criminal | black guy) = p(black guy | criminal) * p(criminal) / p(black guy).
Problem is, the prior (in all the cases I've checked) is lot smaller than the marginal (there's usually at least two orders of magnitude difference!), so even a seemingly high likelihood "washes out" in the posterior.
(In statistics, the likelihood is the probability of the data conditional on the hypothesis, so in this case the probability of being of a certain race given that one is a criminal.)
Let's say we have a group of 1000 animals: 900 rabbits and 100 foxes.
We also know that there are about 10 crimes happening every year and for every crime there's 50% chance of it being committed by a fox.
Then according to Bayes' formula:
P(fox is a criminal) = P(criminal is a fox) * P(animal is a criminal) / P(animal is a fox) = 0.5 * 0.01/0.1 = 0.05 = 5% chance for every given fox to be a criminal.
For every given rabbit the same formula gives 0.5 * 0.01/0.9 which is roughly 0.6%.
Nevertheless, if you want to find a kangaroo in a wild, heading to Australia, would probably be your best bet.
Incels have killed more people than Islamic extremists in the US in the last decade.
Research says drug usage rate is similar for black and white people in the US, but prosecution and conviction rates are not[1]
1. http://www.hamiltonproject.org/charts/rates_of_drug_use_and_...
// Assuming the police are not incompetent, of course, which may not hold in your hypothetical world where they investigate non type X people for terrorism even though they commit zero percent of terrorist attacks - probably out of some misguided diversity quota or affirmative action policy, I assume? ;)
99,999, not 999 innocents and 1:100,000, not 1:1000 (making your point stronger)
> There's a 0.001% chance of finding a terrorist per random sampling.
However, you have to remember that this is an anecdote - you don't want to throw people into boxes they don't fit.
Huh? If 10% of purple people in an area have been incarcerated, and you select one individual from the purple people population of that area at random, the odds are 10% the individual you selected will have been incarcerated. That's fundamental to how sampling works.
Instead, the situation is that 90% of those incarcerated are purple. You can’t draw any meaningful conclusions from that without knowing the number that you were referring to: what percentage of purple people that makes up.
E.g. it may be true that most criminals in an area are of a certain race; that says nothing about the probability of a single person of that race being a criminal. Same with Islamaphobia–even if most acts of terrorism were performed by Muslims (which isn’t even true!), that’d say nothing about the likelihood that a randomly chosen Muslim person you meet is a terrorist.
Edit: changed likelihood to probability