The formula to calculate the binomial probability is:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]
Where:
Let's say we have 10 trials, the probability of success is 0.5, and we want to find the probability of exactly 3 successes. Using the formula:
\[ P(X = 3) = \binom{10}{3} (0.5)^3 (0.5)^{10 - 3} = 120 \times 0.125 \times 0.0078125 = 0.1171875 \]
So, the probability of exactly 3 successes is 0.1171875.
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is used in various fields such as statistics, finance, and science to model binary outcomes.