Bernoulli trials and Binomial distribution

Bernoulli’s Trials are those trials in probability where only two possible outcomes are Success and Failure or True and False. Due to this fact of two possible outcomes, it is also called the Binomial Trial.

Binomial Distribution is the sequence of independent experiments with each experiment being a binomial trial.

1. Bernoulli distribution : Distribution with only 2 outcomes regardless of what outcome is more likely to occur. Example : tossing a coin
2. Binomial distribution : Carrying out bernoulli distribution several times in a row is called as binomial distribution. Which means two outcomes per iteration and many iterations. Example : likelihood of getting 2 heads when the coin is tossed 3 times

Examples of Bernoulli’s Trials

Some of the examples of Bernoulli’s trials are as follows:

• The most common example of the Bernoulli trials is flipping a coin. Each flip of the coin has only two possible outcomes: Heads and Tails. If we consider the Head to be a success, then automatically the tail becomes a failure and vice versa is also true.
• Other than this, rolling a die to get a specific number is also an example of Bernoulli’s Trials. Here if we consider getting a desired number to be a success then any other outcome other than the desired number becomes a failure. In this case, each roll of the dice is a Bernoulli’s Trial.

Difference between Bernoulli trial and binomial distribution

Bernoulli Trial:

• A Bernoulli trial is a random experiment with only two possible outcomes: success or failure.
• The probability of success is denoted by p, and the probability of failure is q=1−p.
• The outcomes are independent, meaning the outcome of one trial does not affect the outcome of another.
• Examples of Bernoulli trials include flipping a coin (heads or tails) or testing the success or failure of a single manufactured item.

Binomial Distribution:

• A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials.
• It is characterized by two parameters: n (the number of trials) and p (the probability of success in each trial).
• The random variable X representing the number of successes follows a binomial distribution and can take values from 0 to n.

Example :

In this game, there are 4 cups where only 1 cup contains a ball. I will shuffle the cups randomly shuffled and have you randomly choose a cup. Thus, the probability of success(or n=1) is

P(n=1) = 1/4 = .25

The expected value of the trial is .25, which is the probability of a success.

The probability of failure(or n=1) is,

P(0) = q = 1-p = .75

Now that a quick example of a Bernoulli Trial is explained, this can be taken a step further. Instead of 1 instance of a game, let there be 5 where the outcomes were [0,0,0,0,1]. The probability of this specific sequence is,

In essence, a binomial distribution is the sum of outcomes from several independent Bernoulli trials, making it a more general and versatile concept.