Binomial Distribution in Statistics Explained with Examples
In the previous section, we listed down some conditions that are to be met for the binomial distribution to be applicable. Let’s take a few examples to understand these conditions in detail.
| Binomial Distribution Applicable | Binomial Distribution Not Applicable |
| Tossing a coin 20 times to see how many tails occur | Tossing a coin until a head occurs |
| Asking 200 randomly selected people if they are older than 21 or not | Asking 200 randomly selected people how old they are |
| Drawing 4 red balls from a bag, putting each ball back after drawing it | Drawing 4 red balls from a bag, not putting each ball back after drawing it |
If you toss a coin 20 times to see how many times you get tails, you are following all the conditions required for a binomial distribution. The total number of trials is fixed (20), and you can only have two outcomes, i.e., tails or heads. The probability of getting a tail is 0.5 each time you toss a coin.
In a way, this is similar to drawing 20 balls out of a bag, replacing each ball after drawing it, and seeing how many of the balls are red. Here, the probability of getting a red ball in one trial is 0.5.
When you toss a coin until you get heads, the total number of trials is not fixed. This is similar to taking out balls from the bag repeatedly until you draw a red ball. You can still find the probability of getting heads in 1 trial, 2 trials, 3 trials etc. and so on, but you cannot use binomial distribution to find that probability.
In the second example, where binomial distribution is not applicable, the experiment does not have only two outcomes, but several. It is similar to taking out balls from a bag that contains red, blue, black, orange, and other-coloured balls. The probability distribution for this experiment cannot be made using binomial distribution.
In the final example, the probability of trials is not equal to each other. For example, the probability of drawing a red ball in the first trial is . Now, in the second trial, the probability of drawing a red ball would be equal to not , as the red ball taken out in the first trial was not put back. Hence, the probability of getting the combination red-red-red-blue, for example, would be ***, which is not the value we got while deriving binomial distribution (***). Again, you cannot use binomial distribution to find the probability in this case.
In other words, binomial distribution is applicable in situations where there are a fixed number of yes or no questions, with the probability of a yes or a no remaining the same for all questions.
So, you now understand that binomial distribution is a very powerful distribution. To get an idea of what this probability distribution looks like, you can use the interactive app provided below. This app shows you the probability distribution for a binomial distribution with n = 5 and p = 0.5. However, you can play around with the values of n and p to see how that changes the probability distribution. Don’t forget to zoom out or zoom in, as needed.
As the professor mentioned, there are some more probability distributions that are commonly seen among discrete random variables. They are not covered in this course, but if you want to go through some of them, you can use the following links: