The binomial distribution deals with two numbers only, these being the probability that an event will happen, p, and the probability that an event will not happen, q. Thus, when a coin is tossed, if p is the probability of the coin landing with a head upwards, q is the probability of the coin landing with a tail upwards. p + q must always be equal to unity. A binomial distribution can be used for finding, say, the probability of getting three heads in seven tosses of the coin, or in industry for determining defect rates as a result of sampling. One way of defining a binomial distribution is as follows:
The binomial expansion of (q + p)n is:
qn + nqn−1p + n(n − 1) 2! q
+ n(n − 1)(n − 2) 3! q
n−3p3 + · · · from Chapter 7.