We first need to understand the Bernoulli Trials to learn about Binomial Probability Distribution.
Bernoulli Trials
Many experiments consist of repeated independent trials and each trial has only two possible outcomes such as head or tail, right or wrong, alive or dead, defective or non-defective, etc. If the probability of each outcome remains the same (constant) throughout the trials, then such trials are called the Bernoulli Trials.
Binomial Probability Distribution
Binomial Probability Distribution is a discrete probability distribution describing the results of an experiment known as the Bernoulli Process. The experiment having n Bernoulli trials is called a Binomial Probability experiment possessing the following four conditions/ assumptions
- The experiment consists of $n$ repeated tasks.
- Each trial results in an outcome that may be classified as success or failure.
- The probability of success denoted by $p$ remains constant from trial to trial.
- The repeated trials are independent.
A Binomial trial can result in a success with probability p and a failure with probability $1-p$ having $n-x$ number of failures, then the probability distribution of Binomial Random Variable, the number of successes in $n$ independent trial is:
\begin{align*}
P(X=x)&=\binom{n}{x} \, p^x \, q^{n-x} \\
&=\frac{n!}{x!(n-x)!}\, p^x \, q^{n-x}
\end{align*}
The Binomial probability distribution is the most widely used in situations of two outcomes. It was discovered by the Swiss mathematician Jakob Bernoulli (1654—1704) whose main work on “the ars Conjectandi” (the art of conjecturing) was published posthumously in Basel in 1713.
Mean of Binomial Distribution: Mean = $\mu = np$
Variance of Binomial Distribution: Variance = $npq$
Standard Deviation of Binomial Distribution: Standard Deviation = $\sqrt{npq}$
Moment Coefficient of Skewness:
\begin{align*}
\beta_1 &= \frac{q-p}{\sqrt{npq}} \\
&= \frac{1-2p}{\sqrt{npq}}
\end{align*}
Moment Coefficient of Kurtosis: $\beta_3 = 3+\frac{1-6pq}{npq}$
Application of Binomial Probability Distribution
- Quality control: In manufacturing, Binomial Probability Distribution can be used to determine the probability of finding a defective product in a batch.
- Medical testing: It can be used to assess the probability of a specific number of positive test results in a group.
- Opinion polls: Binomial Probability Distribution can be used to estimate the margin of error in a poll by considering the probability of getting a certain number of votes for a particular candidate.
By understanding the binomial distribution, you can analyze the probability of success in various scenarios with two possible outcomes.