Probability of a Simple Event and Its Complement

Every outcome either belongs to an event or it doesn't. There's no third option. So if you add up the probability of an event and the probability of everything outside it, you always land on 1, the whole space of outcomes. Write it as P(event) + P(not event) = 1, or P(event) = 1 - P(not event). This becomes a shortcut whenever the 'not' side is a simpler, smaller count than the 'yes' side.

The test loves to hide this shortcut inside 'at least one' questions. 'At least one success in several tries' sounds like it could mean one, two, three, or more successes, which is a mess to count directly. But the complement of 'at least one success' is just one clean outcome: zero successes, meaning failure on every single trial. Find the chance of failing every trial, multiply those failure probabilities together since the trials are independent, then subtract from 1.

This idea connects directly to independence and to counting problems elsewhere in the unit. Any time a problem description includes the phrase 'at least one,' your first move should be to ask what the complement looks like, because it usually collapses a complicated count into a single clean event. The same logic underlies expected value and simulation problems, where you often model the complement to check your work on the original event.