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<p>If your odds (of at least one acceptance) are 1 percent at a one institution, they are essentially quadrupled by applying to four similar institutions. Probabilities are close to being additive in the low range. For applicants who calculate rationally that one longshot is worth it (they are not just sending a lottery ticket to Harvard because having that longshot is a goal in itself), additional longshots are worth it for the same reason.</p>
<p>It is only at the very high probabilities that the odds improvement is marginal, and in those cases, it isn’t the odds that matter; one wants to have multiple offers, not a single acceptance. Conclusion: send more applications.</p>
<p>In the moderate probability range, sending more applications has the greatest chance of making a difference between having some acceptances and having none. Conclusion: send more applications.</p>
<p>This uniformity of strategic implications reflects the fact that expected number of acceptances is a better measure of what applicants are aiming to improve. In all probability ranges, the expected number of acceptances is (crudely) proportional to the number of applications. Double the number of schools, double the expected number of acceptances. </p>
<p>Given this, and the extent to which one might over- or under-estimate the probabilities, for purposes of this question it almost makes no difference to strategy whether one can guess the probabilities correctly. It is hard to think of a scenario where (subject to the obvious non-mathematical externalities) it makes sense to send one application and not a much higher number.</p>