Probability of an Event and Its Complement

Every outcome either happens or it doesn't, so P(A) + P(not A) = 1 always. That one fact is the complement rule, and it matters most when the event you actually want is 'at least one' of something. Counting every way to get at least one success — one six, one defective bulb, one heads — means adding up cases for exactly one, exactly two, exactly three, and so on. That's a lot of bookkeeping. The complement of 'at least one' is just 'none,' and 'none' is one clean case: every single trial has to fail to produce the outcome. So instead of building the event up, you find the probability it never happens at all, then subtract from 1.

The mechanism is the same every time. If trials are independent, P(none) is the product of each trial's probability of failure. Roll a die four times: P(no six on one roll) = 5/6, so P(no six in four rolls) = (5/6)^4. Then P(at least one six) = 1 - (5/6)^4. The independence matters — you're multiplying probabilities because each roll or each draw doesn't affect the others.

This sits right before the binomial setting, because 'at least one success in n independent trials' is really a binomial question in disguise — you're just avoiding the full binomial formula by collapsing all the success counts (1 through n) into their complement (0 successes). Once probability distributions and binomial probability show up, you'll recognize this same 1 - P(X=0) trick as a shortcut built into that bigger model.