Binomial Distribution

A Binomial Distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is defined by two parameters: n (number of trials) and p (probability of success).

In the binomial probability formula, 'n' represents the number of trials or experiments conducted.

In the binomial probability formula, 'p' represents the probability of success on a single trial.

The formula is given by: $$P(X = k) = C(n, k) p^k (1-p)^{n-k}$$ where C(n, k) is the binomial coefficient.

The binomial coefficient C(n, k) is calculated as $$C(n, k) = \frac{n!}{k!(n-k)!}$$ , representing the number of ways to choose k successes in n trials.

The expected value (mean) of a binomial distribution is given by the formula: $$E(X) = n \times p$$ .

The variance of a binomial distribution is given by the formula: $$Var(X) = n \times p \times (1 - p)$$ .

To calculate the probability of at least k successes, use: $$P(X \geq k) = 1 - P(X < k) = 1 - \sum_{i=0}^{k-1} P(X = i)$$ .

Using the binomial formula: $$P(X = 2) = C(20, 2) (0.11)^2 (0.89)^{18} = 0.282$$ .

Using the binomial formula: $$P(X = 3) = C(6, 3) (0.25)^3 (0.75)^{3} = 0.132$$ .