Is your example of the base rate fallacy actually fallacious?

You gave the following example of the base rate fallacy:

Example #1: Only 6% of applicants make it into this school, but my son is brilliant! They are certainly going to accept him!

Explanation: Statistically speaking, there is a 6% chance they will accept him. The school is for brilliant kids, so the fact that her son is brilliant is a necessary condition to be part of the 6% who do make it.

I'm not convinced that this is an example of the base rate fallacy. In fact, I think your explanation might commit a fallacy one might call, "failure to conditionalize". I'll grant you this: "the school is for brilliant kids, so the fact that her son is brilliant is a necessary condition to be part of the 6% who do make it." But from that, it doesn't follow that "statistically speaking, there is a 6% chance they will accept him."

Statistically speaking, we know that the school accepts only 6% of applicants. But we don't know that every applicant to the school is brilliant (lots of dull students apply to dream schools). For example, suppose that only 7% of those who apply are brilliant. In this case, if brilliance is a necessary condition for acceptance, then the woman's son actually has a 6/7 chance of being accepted (or very close to certainty).

The question is whether her son has a very high chance of being accepted, conditional upon his brilliance. That will be significantly more probable than the 6% chance that a random applicant will be accepted, unless most applicants are also brilliant .

Here's an explanation that gets closer to what you have in mind:

Explanation: Statistically speaking, he may still have a low chance of acceptance. The school is for brilliant kids (and everyone knows this), so the vast majority of kids who apply are brilliant. Of these brilliant kids, about 6% get accepted. So her son has a low chance of being accepted (about 6%).

I'm still not convinced that the majority of applicants to an elite school are brilliant. But this at least gets at why the base rate fallacy is fallacious.

Hi Clint,

I see your point here, and perhaps I could have worded this better. I was thinking "brilliance" was defined by standardized test scores in which case the school would have minimum required test scores to apply. All those who apply therefore are "brilliant". From this, it would follow that the woman's brilliant son still has a 6% chance of making it in, all other variables being equal.

As for elite schools only having brilliant kids applying or "brilliance" being defined by standardized test scores, these are hypothetical assertions for the example only and certainly don't reflect my beliefs :)