How does the complement rule help calculate the probability of at least one head?

The complement rule is a fundamental principle in probability theory that simplifies calculating the likelihood of an event by focusing on its opposite. For any event $A$, the complement $A^c$ consists of all outcomes in the sample space that do not belong to $A$. Because the sum of the probabilities of an event and its complement must equal 1, the rule is expressed as $P(A^c)=1-P(A)$.1

When calculating the probability of getting "at least one head" in $n$ tosses of a coin, it is often easier to identify the complement: the event that "no heads" occur. Since there is only one outcome where no heads appear (all tails) out of $2^n$ total possible outcomes, the probability of no heads is $1/2^n$. By applying the complement rule, you subtract this value from 1 to find the probability of at least one head:2

$$P(\text{at least one head})=1-P(\text{no heads})=1-\frac{1}{2^n}$$

This approach avoids the complexity of summing the probabilities of every individual scenario that results in one, two, or more heads.1