Binomial Random Variable

4 min readAug 28, 2019

To understand Binomial Variable it is required to understand what are random variables and types of random variables. If you are not familiar with these- please visit Random Variables and Types Of Random Variables. Assuming you have prior knowledge about these topics, let’s try to understand binomial random variables.

Definition

A binomial random variable counts how often a particular event occurs in a fixed number of tries or trials. Example: Number of heads after 10 flips of a coin. Here the experiment is flipping a coin 10 times and trials represents a single flip. Let’s see what are the conditions for the binomial variable.

Conditions for Binomial Variable:

Trials should be independent of each other.
Each trial can be classified as either a success or a failure.
Fixed number of trials.
Probability of success on each trial should be constant.

Example for binomial variable

Consider a coin which is biased i.e P(Head)=0.6 and P(Tail)=0.4. Let’s have a random variable X which stands for

X = Number of heads after 10 flip of a coin

During this experiment of 10 trials. Each trial(or flip) is independent of any other trial hence this satisfies the condition 1. Now the other condition is each trial can be classified as either a success or a failure. In the context of X we can define ‘head’ as success and ‘tail’ as failure hence each trial clearly has one of the two discrete outcomes. This satisfies condition 2. The number of flips in this experiment is also fixed to 10 and the probabilities are also fixed throughout the experiment. Hence all of the conditions are satisfied.

Variables which are not binomial in nature

Number of kings after taking 2 cards in 10 trials

In order to understand how binomial variables are different let’s see a variable which will not fall under binomial category.

Y = Number of kings after taking 1 card from standard deck without replacement after 10 trials.

Here each trial can be classified as success or failure i.e If we take out a card and it turns out to be a king then it is a success. So during this trial probability of getting a king was 4/52 [Note: we will have 4 kings in the standard deck]. Now let’s perform the second trial. During this experiment probability of getting king will be 3/51 because we have removed a card during our first trial. Hence the second trial is dependent on the first trial. This violates the condition one. Hence Y is not a binomial variable.

If we get rid of “without replacement” in the above statement that is

Y = Number of kings after taking 1 cards from standard deck after 10 trials.

Then this becomes a binomial variable because after every trial we place the card back in the deck hence the number of cards will be constant throughout the experiment.

Binomial Variable Distribution

Consider an example: Let’s say we define a random variable X as the number of heads we get after 3 flips of a fair coin. Since it follows all the above-listed conditions it is a binomial distribution. We can also observe that we can list all the possible outcomes of this experiment. Hence we can say that:

Binomial variable distribution is a discrete variable distribution.

You can refer to this post to understand random variable distributions. Hence we can draw the distribution for this experiment as follows:

Conclusion

In this post, we saw what are binomial random variables and also went through the conditions that are needed to be satisfied by a random variable to be treated as a binomial random variable. We also saw how to draw binomial variable distributions.

Find the next post Divergence in Binomial Variable