At Least One (Complement)

Statement

The probability of getting at least one success in repeated independent trials can be calculated using the complement rule:

\[ P(\text{at least one success}) = 1 - (1-p)^n \]

Here, \(p\) is the probability of success in one trial, and \(n\) is the number of trials.

Why it’s true (short reason)

• It’s often easier to calculate the probability of the opposite event.
• The opposite of “at least one success” is “no successes.”
• The probability of “no successes” is \((1-p)^n\), since every trial must fail.
• Therefore, the probability of at least one success is \(1 - (1-p)^n\).

Recipe (how to use it)

1. Identify the probability of success in one trial, \(p\).
2. Work out the probability of failure: \(1-p\).
3. Raise this to the power \(n\), the number of trials.
4. Subtract from 1 to get the probability of at least one success.

Spotting it

Use this formula when:

• You are asked for the probability of “at least one” success, event, or outcome.
• The trials are independent (outcomes do not affect each other).
• You know the probability of a single success and the total number of trials.

Common pairings

• Binomial probability, where “at least one” is a common case.
• Geometric probability, for waiting times until first success.
• Complementary probability in general (e.g. “at least one head” vs “no heads”).

Mini examples

1. Coin tosses: Toss a fair coin 3 times. Probability of at least one head? \(p=0.5, n=3.\) \(P = 1 - (1-0.5)^3 = 1 - 0.125 = 0.875.\)
2. Dice rolls: Roll a die 4 times. Probability of at least one six? \(p=1/6, n=4.\) \(P = 1 - (5/6)^4 \approx 0.5177.\)

Pitfalls

• Forgetting to subtract from 1.
• Using \(p^n\) instead of \((1-p)^n\) for the complement.
• Applying the formula when trials are not independent.
• Mixing up “at least one” with “exactly one.”

Exam strategy

• Underline the phrase “at least one” in the question.
• Check the independence of events before applying the formula.
• Always calculate failure probability first (\(1-p\)).
• Use a calculator for powers to avoid mistakes with large \(n\).

Summary

The complement rule is a powerful shortcut: instead of adding probabilities for one success, two successes, and so on, you calculate the easier opposite event. With \(P(\text{at least one}) = 1 - (1-p)^n\), you can quickly handle problems involving multiple independent trials.