#### Subject 5. The Binomial Distribution

When a coin is flipped, the outcome is either heads or tails. 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.

A Bernoulli trial is an experiment with two outcomes, which can represent success or failure, up move or down move, or another binary outcome. As one of these two outcomes must definitely occur, that is, they are exhaustive, and also mutually exclusive, it follows immediately that the sum of the probabilities of a "success" and a "failure" is 1.

A binomial random variable X is defined as the number of successes in n Bernoulli trials. The assumptions are:

• The probability (p) of success is constant for all trials. Similarly, the failure probability 1 - p stays constant throughout the experiment.
• The trials are independent. Thus, the outcome of one trial does not in any way affect the outcome of any subsequent trial.
• The sampling is done with replacement. This means that once an outcome has occurred, it is not precluded from occurring again.

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 p is the probability of success on any one trial. This formula assumes that the events are:

• dichotomous (fall into only two categories)
• mutually exclusive
• independent
• randomly selected

To remember the formula, note that there are three components:

• n!/[(n-r)! x r!]. This indicates the number of ways r successes can be achieved and n - r failures in n trials, where the order of success or failure does not matter. This is the combination formula.
• pr. This is the probability of getting r consecutive success.
• (1 - p)n-r. This is the probability of getting n - r consecutive failures.

The values for n and p will always be given to you in a question; their values will never have to be guessed.

Consider this simple application of the binomial distribution. What is the probability of obtaining exactly 3 heads if a coin is flipped 6 times? For this problem, n = 6, r = 3, and p = 0.5, =>p(3) = {6!/[(6 - 3)! x 3!]}0.53(1 - 0.5)6-3 = 0.3125.

Often the cumulative form of the binomial distribution is used. To determine the probability of obtaining 3 or more successes with n = 6 and p = 0.3, compute p(3) + p(4) + p(5) + p(6).

For a single Bernoulli random variable, Y, which takes on the value 1 with probability p and the value 0 with probability 1 - p, the mean is p and the variance is p(1 - p).

Every random variable has a mean and a variance associated with it. A general binomial random variable, B(n, p), is the sum of n Bernoulli random variables, and so the mean of a B(n, p) random variable is np. Given that a B(1, p) variable has variance p(1 - p), the variance of a B(n, p) random variable is n times that value, or np(1 - p), using the independent assumption.

For example, for a B(n = 5, p = 0.10) random variable, the expected number of successes is 5 x 0.1 = 0.5 with a standard deviation of (5 x 0.1 x 0.9)1/2 = 0.67.

A tracking error is a measure of how closely a portfolio follows the index to which it is benchmarked: total return on the portfolio - total return on the benchmark index.