Introduction
A discrete random variable $x$ is said to have a binomial distribution if $x$ (binomial random variable) satisfies the following conditions:
Table of Contents
- An experiment is repeated for a fixed number of trials $n$.
- All the trials of the experiments are independent of each other.
- All possible outcomes for each trial of the experiment can be classified into two mutually (complementary) events: one is $S$ (called success) and the other is $F$ (called failure).
- The probability of success $P(S)$ has a constant value of $p$ for every trial (that is, the probability of success is fixed for each trial) and hence the probability of failure $P(F)$ has a constant/fixed value of $q$ for every trial, where $q=1-p$.
- The random variable $x$ counts the number of trials on which $S$ (success) occurred.
Calculating Probabilities for a Binomial Random Variable
If $X$ is a binomial random variable with $n$ trials, probability of success $p$ (and probability of failure $q$), then by the fundamental counting principle, the probability of any outcome in which there are $x$ successes (and therefore $n-x$ failures) is
To count the number of outcomes with $x$ successes and $n-x$ failures, one can observe that the $x$ successes could occur on any $x$ of the $n$ trials. The number of ways of choosing/selecting $x$ trials out of $n$ is $\binom{n}{x}$, so the probability of $x$ successes becomes:
$$P(X=x)=\binom{n}{x} p^x q^{n-x}$$
Example of Binomial Random Experiments
Example: Consider the experiment of flipping a coin 5 times. Let the event of getting Tails on a flip is considered a “success”. Also, suppose that the random variable $T$ is the number of tails obtained, the $T$ will be binomially distribution with $n=5, p=\frac{1}{2}$, and $q=\frac{1}{2}$.
Solution:
Suppose the random variable $T$ represents the number of trials when a coin is flipped three times.
$$P(X=2) = \binom{3}{2}\left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 = 0.375$$
Properties of Binomial Distributions
In many cases, one may be interested in the mean and standard deviation of the binomial random variable. If $x$ is a binomial random variable with $n$ trials with probability of success $p$ and probability of failure $q$, then the mean and standard deviation of $x$ can be computed as
- Mean: $E(X) = \mu(x) = np$
- Standard Deviation: $\sigma(x) = \sqrt{npq}$
- Variance: $npq$
Note that
- A binomial distribution is symmetric if $p=q$,
- left skewed if $p>q$ and
- right-skewed if $p<q$