When a coin is flipped, the outcome is either a head or a tail; when a magician guesses the card selected from a deck, the magician can either be correct or incorrect; when a baby is born, the baby is either born in the month of March or is not. In each of these examples, an event has two mutually exclusive possible outcomes. For convenience, one of the outcomes can be labeled "success" and the other outcome "failure." If an event occurs N times (for example, a coin is flipped N times), then the binomial distribution can be used to determine the probability of obtaining exactly r successes in the N outcomes. The binomial probability for obtaining r successes in N trials is:

where P(r) is the probability of exactly r successes, N is the number of events, and π is the probability of success on any one trial. This formula for the binomial distribution assumes that the events:

- are dichotomous (fall into only two categories)
- are mutually exclusive
- are independent and
- are randomly selected

Free Tutorial on Using Excel's Binomial Function