Cue Column:

• When to use complement?
• Formula for "at least one"?
• When is complement faster?

Note Column: Use complement when computing P(at least one) directly is complex. P(at least one) = 1 - P(none). For n independent events each with failure probability $q_i$: P(at least one success) = 1 - $q_1$ * $q_2$ * ... * $q_n$. If all have the same failure probability q: P(at least one) = 1 - $q^n$. Similarly, P(X >= 2) = 1 - P(X=0) - P(X=1) is faster than summing P(X=2) + P(X=3) + ... + P(X=n).

Summary: The complement approach converts a multi-case calculation into a 1- or 2-case calculation. Always consider it for "at least" problems.