Validity Of Binomial Distribution

Binomial distribution: discrete probability distribution of the number of successes in a sequence of $n$ independent yes/no experiments, each of which yields success with probability $p$

Null hypothesis: there is no there is no significant difference between specified populations, any observed difference being due to sampling or experimental error

$$P(X = k) = \binom n k p^k(1-p)^{n-k}$$
$$P(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n\choose i}p^i(1-p)^{n-i}$$