We have seen that we can calculate the probability of observing a specific value using a probability mass function. What if we want to find the probability of observing a range of values for a discrete random variable? One way we could do this is by adding up the probability of each value.

For example, let’s say we flip a fair coin 5 times, and want to know the probability of getting between 1 and 3 heads. We can visualize this scenario with the probability mass function:

We can calculate this using the following equation where P(x) is the probability of observing the number x successes (heads in this case):

$\begin{aligned}
P(1\;to\;3\;heads) = P(1 <= X <= 3) \\
P(1\;to\;3\;heads) = P(X=1) + P(X=2) + P(X=3) \\
P(1\;to\;3\;heads) = 0.1562 + 0.3125 + 0.3125 \\
P(1\;to\;3\;heads) = 0.7812 
\end{aligned}$