Explain how the complement can be used to find the probability of getting at least one item of a particular type

The outcome A="At least one" means {1,2,3,4,5 ... n} where n is the maximum possible outcome.

The only other outcome (B) left (i. e. the complement of the above event "at least one") is (B) = {0}.

Therefore, by Kolmogorov's second axiom, A and B are complements, and

P (A) + P (B) = 1

therefore

P (A) = 1-P (B), or in words,

Probability of outcomes of "at least one" is the complement of the outcome of "0" item.