Understanding Binomial Distribution with Example

In this class, we discuss Understanding Binomial Distribution with Example.

For Complete YouTube Video: Click Here

The reader should have prior knowledge of Bernoulli distribution. Click Here.

Binomial Distribution: A random experiment consisting of n repeated Bernoulli experiments.

1) Each experiment is independent

2) Each experiment results in success or failure.

3) The probability of success in each experiment is p.

We can apply binomial distribution if the random experiment follows the above conditions.

We take an example and understand binomial distribution.

A random variable X that results in the number of experiments that result in success has a binomial distribution with parameters n and p.

Probability mass function = P(x; n , p) = nCx (p)^x (1-p)^n-x

Example: Toss a coin six times.

Probability of having two successes?

Success = head.

One way: H, T, T, T, T, H

The first experiment and last experiment showed head.

The remaining experiments showed tails.

Probability = P 1-p 1-p 1-p 1-p p. because p is success and 1-p is a probability of failure.

The probability = (p)^2 (1-p)^6-2. because all experiments are independent.

From the multiplication theorem, independent events are the multiplication of each value.

Second Way: H, H, T, T, T, T.

Similarly, we can have nCx possibilities.

So probability mass function = nCx (p)^x (1-p)^n-x

The below table shows the binomial distribution for the experiment tossing a coin six times.

When we toss a coin six times. The random variable X takes values 0, 1, 2, 3, 4, 5, 6.

We can have zero heads, one head, etc.

The probability P(X= 0) = f(x) = nCx (p)^x (1-p)^n-x where n = 6 and x = 0.

Given, The probability values for random variable X in the table.

The binomial distribution follows the discrete probability distribution.

The discrete probability distribution should follow two conditions.

1) Σall x (f(x)) = 1

2) f(x) >= 0