For example, if you have a $1.5\%$ chance of obtaining admission to any school and you apply to $15$ schools what is the chance that you'll get into a least $1$ school? Is this as simple as $1.5\%$ $×$ $15 $= $22.5\%$ chance of getting into one school?
Phrased another way: if an event has a $1.5\%$ chance of occurring every time it is performed, and you perform it $15$ times what is the probability that the event occurs at least once?
Best Answer
If an event happens with probability $p$, then the probability of it not happening is $1-p$. Instead of looking at the question "does the event occur at least once?", consider its complement. That is, "what is the probability of the event never occurring?"
If we repeat the event $n$ times, the probability of the event never happening is $(1-p)^n$. Thus, the probability of it happening at least once is $1-(1-p)^n$.
Applied to your scenario, we have $p=0.015$, $n=15$. Thus...
The reason that we can't just multiply the probability of one occurrence by the number of samples is best illustrated by an example: Suppose we had $p=1.5\%$, as in your problem. If we applied to $1000$ schools, what is the probability that we get in? Saying $P[\text{I get in}] = 1000\cdot 1.5\% = 1500\%$ doesn't make since--probabilities must be between $0$ and $1$!
EDIT: I wrote this up, and then saw André's comment on the main post. The assumption of independence is key, here. If the probability of the event changes every time you perform an experiment, the answer is a bit more involved.