There is an element of chance in getting into any top school, but only among the applicants who meet all the other qualifications. That is, between two students with 4.0 GPA, great SAT scores, and a wide selection of extracurricular activities and community service, it's a toss up. But a student with poor grades has no chance. It doesn't matter how many times the student with bad grades sends his application, he isn't getting in just by random chance. When it comes to Harvard, I would guess that 80-90% of applicants can be dropped immediately due to grades, SAT scores or lack of activities. Out of the remaining pool, for those without extenuating circumstances such as athleticism or family connections, is where such as race and "personality" might become a factor, and who ends up getting picked has an element of randomness.

Fair! I did look only at the 'non-legacy/athletic' Harvard acceptance rates (hard to find and piece together, btw), so we're looking at a bit more apples to apples comparison, though, as you say, not totally.

Lets say that only 10% of the applicants that enter can win. If you think that you are a possible winner in terms of scores, grades, etc. then what should you pay to enter the lotto? Please correct my math if I am wrong (probability is always a mind warp). The further reduction in probability to win makes the nominal acceptance rate not 3%, but 0.3%. That means that the price of the hypothetical lotto ticket should be $8,980. You just move the decimal over.

Still, compared to the $75 application fee, this is a steal. The acceptance rate would need to be ~0.003% to get the lotto ticket price to be about what the application fee is. That means Harvard would need to be ~1000x more picky than they currently are to get the lotto priced correctly.

Harvard had ~40k applications last year. At 1:1000 odds being considered 'worthwhile' due to the hypothetical lotto ticket cost, then you should apply if you think you are better than the bottom 40 applicants. You can play with the probabilities and make up a lot of hypothetical curves yourself. Nonetheless, the math seems to indicate that the $75 application fee is really low if you think you have nearly any chance of getting in and the process has nearly any randomness to it.

> Based on last year's acceptance rate of ~3% and the expected lifetime earnings of ~$3,000,000, then the price you should pay for that lotto ticket should be $89,800.

It's not the lifetime earnings from a Harvard education that are relevant, it's the increase of lifetime earnings at Harvard relative to a school one is effectively guaranteed to get into (say your state school that uses objective criteria).

That number is pretty difficult to evaluate, but I'm confident it is well under $1M.

> . Now, compare that to the actual price of ~$75 for an application fee. Even a flimsy application to Harvard is totally worth the money.

The cost of applying to an additional college is generally in time, not money. The money is just there to discourage low-quality shotgunning.

Nit: you need to calculate the Net Present Value of that $3,000,000, as well as subtracting the cost of tuition.

True! That number is, at best, a lower bound. Interest rates will likely make it MUCH larger, even with an inflation factored in. However, I've no way to calculate that as I don't have a crystal ball.

Good catch on the tuition! I did subtract it but did not mention it as I thought it would not read as well. In the end, the price of the lotto ticket should be MUCH higher when compound interest is factored in. How much is anyone's guess.