Why does an endlessly tossed biased coin eventually produce a head with probability one?

In the context of probability theory, the assertion that an endlessly tossed biased coin will eventually produce a head with probability one is a consequence of the properties of independent events and infinite series.

If a coin has a probability $p>0$ of landing on heads in a single trial, the probability of failing to get a head in $n$ independent tosses is $(1-p{)}^n$. As the number of tosses $n$ approaches infinity, the probability of never observing a head is:

$${lim}_{n\to \infty }(1-p{)}^n=0$$

Because the probability of the complement event (never getting a head) is zero, the probability of the event occurring at least once is $1-0=1$. This result is consistent with the geometric distribution, which models the number of trials required to achieve the first success in a sequence of independent Bernoulli trials. As long as the probability of success $p$ is greater than zero, the probability of failing indefinitely is zero, ensuring that a head will eventually appear with probability one.1