I'm a bit confused about why frequentist measures of probability based on groups are relevant to individual cases. It seems that moving from the group to the individual is somehow a violation of the fallacy of composition or is a hasty generalization.

Let's say there is a frequency where 10% of the attending students successfully pass a specific test in mathematics and 90% fail. Linda tells you that she has passed the test, but you do not have any evidence except her own and a friend's testimony that she attended and did pass. If Linda has passed, she is entitled to get 1 million USD from her grandfather. And no presumption can be made about the likelihood of Linda lying about passing.

How logically is the proposition of the 10% pass rate among students related to the assessment of Linda's individual testimonial claim? Perhaps more precisely what is the logic or argument that can be applied regarding frequentist probabilities?

  • 2
    David Lewis's principal principle might be of interest to you. "The Principal Principle says that a rational agent conforms their credences to the chances." Of course here we have additional information, namely whatever we know about Linda in particular, and how reliable Linda's testimony is, so we conditionalize on those piece of information too. If Linda's a good student, and she rarely lies then it should be very likely that she passed. Dec 5 '20 at 11:35
  • Many Thanks Adam! I've started to study probability of causation as well. Dec 7 '20 at 19:57
  • Hi, welcome to Phil.SE! Can you cite, or provide more information, on the theory you present as popular? Sure we learn from probability to singular cases, but this is a bit extreme with many parameters which you claim to not be considered in the statistics. And if the question ends up being simply about theory of probability, it probably doesn't belong in this SE but rather on Math or Statistics. Dec 29 '20 at 20:53
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    – J D
    Jan 3 at 15:52
  • Edited in response to multiple closure votes.
    – J D
    Jan 3 at 16:23

Yes, you cannot use this base pass rate of 10% to effectively judge whether Linda passed the test, because Linda is in a particular situation, with the $1 million reward, that makes her different from the typical student taking the test. We would expect that the reward would make her highly motivated to pass the test, so she would be more likely to pass. But she may also have a stronger incentive to lie about having passed. It would be necessary to use a Bayesian approach to estimate the chance that she passed (versus the chance that she lied).

Really, even under the frequentist interpretation, probability is about an agent's lack of knowledge. For example, if you are told that a die was rolled, the probability it came up 5 is 1/6. If you are told a die was rolled and it did not come up 3, the probability it came up 5 becomes 1/5, because you gained knowledge. The frequentist interpretation and the Bayesian interpretation give the same result here. As probabilities are about what an agent knows or doesn't know, probabilities are subjective, even with the frequentist interpretation.

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