Binomial Distribution

When an experiment involves a ``trial'' such that
  1. it results in possibly two outcomes, say ``success'' or ``failure,'' and
  2. it is repeatedly and independently performed (that is, each trial is designed to be identical but does not interfere with others),
it is called a binomial experiment. The probability of success remains the same probability $ p =$ in each trial, and a trial is performed repeatedly $ n =$ times.

Tossing a coin is an example of independent trial where head (red face) or tail (blue face) are the two possible outcomes. When it is tossed repeatedly, the experiment yields the number of successes (red faces).

In the experiment the number of successes is the random variable of interest. From the probabilistic point of view it is randomly drawn from the common probability distribution. (Here it is computationally limited to $ n \le 50$.)

The binomial experiment itself can be reproduced repeatedly, say times. Then the outcome recorded every time reveals the shape of frequency distribution. The exact frequency function $ f(k) = P(X = k)$ is formulated as

$\displaystyle f(k) = {}_{n}C_{k}  p^k (1-p)^{n-k}$    for $ k = 0,1,\ldots,n$,

and it is called a binomial distribution with parameters $ n$ and $ p$. Here $ {}_{n}C_{k}$ is the number of combinations of choosing $ k$ items from $ n$ items.