How does the complement rule find the probability of at least one head in repeated coin tosses?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, the complement rule is a powerful tool for calculating the probability of an event by subtracting the probability of its opposite (the complement) from 1.

To find the probability of getting "at least one head" in $n$ coin tosses, it is mathematically simpler to first calculate the probability of the complement event: "no heads occur" (which is equivalent to getting all tails).

Identify the total outcomes: For $n$ tosses, there are $2^{n}$ possible outcomes in the sample space.
Find the complement probability: There is only one outcome where no heads occur (all tails). Thus, the probability of no heads is $1 / 2^{n}$ .
Apply the complement rule: Since the sum of the probabilities of an event and its complement must equal 1, the probability of at least one head is:
$1 - \frac{1}{2^{n}}$

This method avoids the need to sum the individual probabilities of getting exactly one head, two heads, and so on, up to $n$ heads.1