The binomial distribution is one of the basic mathematical models for describing the behavior of a random outcome. An experiment is performed that results in one of two complementary outcomes, usually referred to as "success" or "failure". Each experimental outcome occurs independently of the others, and every experiment has the same probability of failure or success.
A toss of a coin is a good example. The coin tosses are independent of each other, and the probability of heads or tails is constant. The coin is said to be fair if there is an equal probability of heads and tails on each toss. A biased coin will favor one outcome over the other. If I toss a coin (whether fair or biased) several times, can I anticipate the numbers of head and tails that can be expected? There is no way of knowing with certainty, of course, but some outcomes will be more likely than others. The binomial distribution allows us to calculate a probability for every possible outcome. Consider another example. Suppose I know from experience that, when driving through a certain intersection, I will have to stop for traffic light 80% of the time. Each time I pass, whether or not I have to stop is independent of all the previous time. The 80% rate never varies. It does not depend on the time of day, the direction that I am traveling, or the amount of traffic on the road.
If I pass through this same intersection eight times in one week, how many times should I expect to have to stop for the light? What is the probability that I will have to stop exactly six times in the eight times that I pass this intersection? The binomial model is a convenient mathematical tool to help explain these types of experiences. Suppose there are N independent events, where N takes on a positive integer value 1,2.......Each experiment either result in success with probability p else results in a failure with probability 1-p for some value of p between 0 and 1.