Def:          Bernoulli trials ≡ repeated identical experiments with independent outcomes that are
                            1 (success) or 0 (failure)

Def:          p ≡ P(1) ≡ probability of success

Def:          q ≡ P(0) ≡ probability of failure = 1 − p

Ex:            Flipping a fair coin constitutes a Bernoulli trial. We may define Heads as success, P(Heads) = P(1) = p = 0.5, and Tails as failure, P(Tails) = P(0) = q = 1 − p = 0.5.

Def:          Binomial distribution ≡ P(m successes in n trials) = nCm⋅pmqn−m =

Note:       The binomial distribution is an example of combinatoric probabilities where the probability of a single outcome is pmqn−m.

Ex:            Suppose p = P(1) = 0.4 for a stream of bits in a communication system. Find the probability of 4 out of 6 bits being 1's.

Sol'n:       There are 6C4 patterns of 6 bits with four bits = 1. The patterns are 001111, 010111, 011011, ... , 111100.

The probability of a particular one of these patterns occurring as the outcome is p4q2. All of the patterns have the same probability, however, so our answer is given by the binomial distribution: