Let me offer the explanation that hopefully can unify the different perceptions people have.
@hzhao2004, @Postmodern, first, I do believe about 30% of 2400 SATers can get into Harvard. I remember quotes where 2/3 or 3/4 of those are rejected? I suppose the rest get accepted.
However, the key issue of using 30% in the way that you guys used is that the population of the 2400 SATers is not a homogenous group. That @mikemac had already mentioned in the other thread. Thus some of those may have 2400 and be STS finalist. Their probability can be 80%+. Some may have 2400 as the only calling card. Their probability may be 5% or less. That’s the hidden information (let’s denote with A) that the number 30% does not comprehend. Now let B be the event of a particular person in the group getting into Yale and C be the event of getting into Harvard. Then B, C will be correlated because both B and C will be affected by A (whether the applicant is “STS” or “nobody”) to some extent.
Now there is actually way to make B and C to be completely independent in the statistical analysis. How? By removing the effect of A. That is, considering only the sub population which is completely homogenous which means everyone in the population has completely the same credential. Of course then admission can’t differentiate between you and your twins and the decision will be random and independent between B and C. But the probablity of occurance for the smaller pool could be drastically different too from the 30% of the bigger pool.
So as a summary, the only time the independence can be used in this analysis is when the pool consists of “identical” people.