e-Statistics

Binomial Distribution

When an experiment involves a “trial” such that

it results in possibly two outcomes, say “success” or “failure,” and
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.

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 exact frequency function f(k) is formulated by

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

and it is called a binomial distribution with parameters n and p. Here $\binom{n}{k}$ is the binomial coefficient, which represents the number of combinations of choosing k items from n items.

	Outcome	Probability
1

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).

The binomial experiment itself can be reproduced repeatedly, say times. Then the outcome recorded every time reveals the shape of frequency distribution.