At Least One Head Probability: The Complement Method

What does at least one head mean?

At least one head means one or more heads. For n independent tosses of a fair coin, its probability is 1 − (1/2)ⁿ. The complement is no heads, which means every toss is a tail. This formula needs both fairness and independence.

Example: three coin tosses

There are 2³ = 8 equally likely ordered outcomes. Only TTT has no head. Therefore P(at least one head) = 1 − 1/8 = 7/8. The favourable outcomes include HHH and outcomes with two heads as well as those with exactly one.

For exactly one head, the outcomes are HTT, THT and TTH, giving 3/8. That is why exactly one and at least one cannot be used interchangeably. Listing the small sample space is a useful independent check on the complement method.

Why the outcomes are ordered

HTT and THT record a head on different tosses, so they are different outcomes. A list containing only zero heads, one head, two heads and three heads has four categories, but those categories are not equally likely. Counting categories as though each had probability 1/4 is the common error.

What changes for a biased coin?

If each independent toss has head probability p, the chance of no heads in n tosses is (1 − p)ⁿ. Thus the required probability is 1 − (1 − p)ⁿ. For p = 0.6 and two tosses, this is 1 − 0.4² = 0.84. If tosses are not independent, multiplying probabilities this way needs additional information.

Practice

1. At least one head in two independent fair tosses: 3/4. Only TT fails.
2. At least one head in four independent fair tosses: 15/16. Only TTTT fails.
3. Exactly one head in two fair tosses: 1/2, from HT and TH.

Before calculating, underline the words at least, exactly or at most. They define different events even when the experiment is identical.

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