Often one will test a critical system components for failure and toward that end collect a sample of 100 of these components from the manufacturer. Suppose the component is listed as having a p = 0.01 probability of breaking and you want to know the likelihood that at most 1 of the tested components actually fails when tested. You find it reasonable to presume that different components succeed or fail independently of each other. So, you can model this situation with a binomial distribution.
If \(X\) measures the number of components that fail when tested, the specific probability function is given by
\begin{equation*}
f(x) = \binom{100}{x} 0.01^x 0.99^{100-x}.
\end{equation*}
The probability that at most one component fails is then given by
\begin{equation*}
F(1) = f(0) + f(1) = \binom{100}{0} 0.01^0 0.99^{100} + \binom{100}{1} 0.01^1 0.99^99 \\ = 0.99^{100} + 100 \cdot 0.01 \cdot 0.99^{99} = 0.99^{99}(0.99 + 100 \cdot 0.01 \\ = 0.99^{99} \cdot 1.99 \approx 0.73576.
\end{equation*}