I’m not assuming that the decisions are somehow dependent. This is just how probability works. To use your coin example, let’s say it’s a fair coin - a heads means you get in, tails means rejection. The probability of getting 1 heads within 5 flips is 1-0.5^5 = 96.9%. Probably an easier way to see this is that the only scenario in which you don’t get at least 1 heads is if you get 5 tails in a row. The probability of that is 0.5^5, or 3.1%.
Similarly, the only way my daughter wouldn’t get into one of the 4 reach schools is if she gets tails on all 4 flips. Of course it’s not a fair coin, so she has a 95% chance of getting a tail on any given flip. The probability of getting 4 tails is 0.95^4, or 81.5%, meaning the probability of getting at least one hit is the remaining 18.5%.
The bigger question is: given my D’s overall profile, are those probabilities roughly correct? For example, if she had a 2.0 average and 1000 SAT score, and I was using 5% as the probability for schools like Brown and Yale, my analysis would be far from accurate.