How does the complement rule help calculate the chance of at least one event?

“At least one” includes one occurrence and every larger number of occurrences. Instead of adding the probabilities for all those possibilities, ask the simpler opposite question: what is the chance it happens zero times? Zero occurrences is the complement of one or more occurrences, so the probability you want is 1 minus the probability of none. [S1]

In symbols: P(at least one) = 1 − P(none). This is especially handy when it is easier to describe or calculate “none” than to list every way the event could happen. The key is to define the event and the trials clearly before calculating.

Suppose, as a mathematical example, you roll a fair six-sided die three times and want the chance of getting at least one four. Assume the rolls are independent and each roll has the same chance of being a four. On one roll, the chance of not getting a four is 5/6. For all three rolls to have no four, multiply those no-four chances: (5/6) × (5/6) × (5/6) = (5/6)³. Then subtract from 1:

P(at least one four) = 1 − (5/6)³ = 91/216, about 0.421.

So the probability is 91/216, or about 42.1%, under those assumptions. This is a probability calculation for the stated die-and-roll setup, not a forecast about a real-world event. The example and complement formula are given by OpenStax. [S1]

Not automatically. In the die example, multiplication works because the setup assumes independent rolls, and each roll has the same 5/6 chance of not being a four. [S1] If trials affect one another, or their chances differ, you cannot simply raise one no-event probability to a power. You need to work out the probability of no event for the actual setup, using its conditions. The complement rule still applies—P(at least one) = 1 − P(none)—but the shortcut for calculating P(none) may not.

Write down the event in plain language first, then its complement. For repeated, independent trials with an equal event chance, use this copyable setup:

Event of interest: [what counts as one occurrence] Number of trials: [n] Chance of the event on each trial: [p] Chance of no event on each trial: [1 − p] Assumption: [why trials are independent and have the same chance] P(none) = [1 − p]ⁿ P(at least one) = 1 − [1 − p]ⁿ

For a different setup, replace the power calculation with a justified calculation of P(none). Before giving a result, check that “none” really is the opposite of the event you want and that your probability model matches the stated assumptions.