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Quite interesting! I would note though that one thing you forgot to consider is where that 5% acceptance probability comes from. I think we could characterize it as the applicant’s guess as to the chance that (for example) Harvard or Yale or Stanford will accept him.

So when we take into account Harvard’s rejection in his chances of being accepted at Yale, what we are actually doing is revising downward the true chance of acceptance, versus the original guess based on the little information the applicant had. The 5% was a guess, and we are now revising that probability downwards based on the new information, each rejection or acceptance bringing it closer to its true figure.

So your math might be correct… but I’m not sure how closely it correlates to what you’re trying to measure. I think you are using the 5% as a putatively correct estimate that any single college on his list will accept him independently of the others. But there is no reason to think that the person estimating this probability is not actually already taking the correlation into account in this 5% figure. Wouldn’t you assume acceptance to Yale and Harvard is strongly correlated?

It’s quite possible that people on average underestimate the effect of the correlation, and are thus getting their 5% figure wrong. But I’m not sure your math, though well executed, actually proves what you’re trying to prove. Very interesting article, though!

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