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Let's look at admissions to top PhD programs in English, where it seems like the top 10 schools each receive about 300 applications for an average of about 6 spots (these are approximations only). Across this sampling of schools, assume that there are 600 unique applicants. If all schools select different candidates, which admittedly is very likely not the case, the chances of being admitted to a top 10 program bounce from something on the order of 2% to more like 10%.

How overly optimistic is this calculation?

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    Interesting question, but I'm having trouble with the premise. Super-qualified people have a relatively high chance of admission; unqualified people have ~0% chances.I suppose you're asking what the odds are for an "average" applicant, but an "average" applicant by definition won't be admitted to a program with a < 50% admission rate (even if there were such a thing as an average applicant).
    – cag51 ♦
    Commented Mar 29, 2019 at 2:07

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I think the basic idea of the calculation is fine. It's actually irrelevant that some superstars will get accepted to multiple programs. The schools know that and compensate with mechanisms such as (1) sending out more acceptances than slots, knowing some percentage will decline, (2) waiting lists, late admission decisions, and early admission encouragement to fill their remaining slots.

So from a practical standpoint, looking at the entire sum of slots filled versus applicants makes sense. Of course details may be slightly more complicated. But the basic insight applies.

Edit: Yes, cag51 is right that your qualifications matter. This is not just picking colored balls out of bags. However, even here the 10% versus 2% point makes sense. (Consider that if a perfect selection process is run, you need to be in the top 10%, not the top 2%.)

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