At Least One Probability vs Complement

At least one and none are complements, so $P(\text{at least one}) = 1 - P(\text{none})$ and you only ever compute one product. Roll a fair six-sided die 4 times: the chance of no 6 on a single roll is $5/6$, and because the rolls are independent the chance of no 6 on all four is $(5/6)^4 = 625/1296 \approx 0.4823$. So the probability of at least one 6 is $1 - 0.4823 = 0.5177$. The power step needs the trials to be independent with the same success probability on each one.

The complement of an event $A$, written $A^c$ (read A complement), collects all outcomes in the sample space where $A$ does not happen. Its probability is $P(A^c) = 1 - P(A)$, since an event and its complement together cover everything. For example, if the probability of rain is 0.3, the probability of no rain is $1 - 0.3 = 0.7$. The complement rule is handy when at least one is easier to compute as 1 minus none.