109 LearnersLast updated on June 18th, 2025
Bernoulli Distribution
An important concept in statistics and data science, the Bernoulli distribution is one of the ways to describe an event with only two possible outcomes. The outcomes are represented as ‘success’ or ‘failure’ or ‘1’ and ‘0.’ Let us understand the concept with an example. Suppose we participated in a race, we either win or lose. The Bernoulli distribution helps explain situations like these. In this topic, we will explore the concept of Bernoulli distribution.
Bernoulli Distribution Definition
A discrete probability distribution that has only two possible outcomes is called Bernoulli distribution. Any random experiment resulting in either 0 or 1 is known as a Bernoulli trial. If the result is 1, it represents success. Also, the probability of success is denoted by ‘p.’
If the result is 0, it means failure and it is denoted by ‘q’ or ‘1-p.’ To see how two possible outcomes work, think about flipping a coin, a team playing a ball game, answering a true or false question, determining if a delivery happens on time, and so on.
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Bernoulli Distribution Formula
The Bernoulli distribution formula expresses the probability of two possible results. It shows the chances of success and failure. The Bernoulli distribution has two important functions, probability distribution function (PDF) and cumulative distribution function (CDF). The PDF gives the probability of success and failure. The CDF, on the other hand, describes the probability of getting a value less than or equal to 1 or 0.
The PDF for Bernoulli distribution is:
P (X =x) = px (1 - p)1 - x
Here,
P (X =x) = The chance of getting the possible outcome, like success or failure.
X = The outcome
Where x = 1 means success and x = 0 means failure
p = Probability of success
1 - p = Probability of failure
Now, let us take a closer look at this formula with an example:
In a running competition, if the probability of winning is 0.7, then the probability of losing is:
1 - 0.7 = 0.3
By using the Bernoulli distribution formula we can find the probability.
Probability of success:
P (X = x) = px (1 -p)1 - x
P (X = 1) = (0.7)1 (0.3)1 - 1
0.7 × 1 = 0.7
So, the probability of winning is 0.7
Probability of failure:
P (X = x) = px (1 -p)1 - x
P (X = 0) = (0.7)0 (0.3)1 - 0
1 × 0.3 = 0.3
Therefore, the probability of losing is 0.3
The cumulative distribution function for Bernoulli distribution is:
FX (x) =
Here, when x < 0 the probability is 0.
If x is between 0 and 1, the probability is 1-p
If x ≥ 1, the probability is 1.
FX (x) = The cumulative distribution function, the random variable X is less than or equal to x.
P (X ≤ x) = Probability of X is less than or equal to x.
Properties of Bernoulli Distribution
Learning the properties of Bernoulli distribution helps students to understand the situations with only two possible outcomes. It is a simple way to know how probability works in real life and also in subjects such as economics, mathematics, statistics, and computer science. The properties of Bernoulli distribution are listed below:
Two possible outcomes:The outcome is always binary, which means they are either success or failure.
While success is represented as 1, failure is denoted as 0.
Probability of success and failure: The probability of success can be denoted as P (X = 1) =p. Whereas, the probability of failure is denoted as P (X = 0) = 1 - p. Also, the probability of success for each and every trial remains the same. For e.g., whenever a coin is flipped in the air, the probability of heads (1) is 0.5. This probability does not change irrespective of the number of times the coin is flipped.
Complementary probability: ‘P’ represents the probability of success and ‘q’ denotes the probability of failure. Here, q = 1 - p because the probability of failure should be whatever is left over from 1.
Independent trials: The outcomes of Bernoulli trials do not affect the result of another. Each trial and its result is independent and does not influence the future outcome. However, the probability of success and failure will be the same.
Expected value(Mean): It is equal to the probability of success. The mean explains the average outcome over many trials.
E (X) = p
This is the formula for calculating the mean or the expected value.
Variance: Variance measures how much the outcomes deviate from the mean. The formula for variance is:
Variance = p (1 - p) = pq
When the variance is high, there is a huge gap between the outcome and the mean. If the variance is low, then the outcomes are closer to the expected value.
Real-life Applications of the Bernoulli Distribution
The concept of Bernoulli distribution can be applied to predict the outcome of events. Probability plays a major role even in day-to-day lives. Here are some examples where it is used.
- In logistics, the Bernoulli distribution helps predict whether an event occurs. For example, it can be used when a delivery company or its clients want to know whether their package can arrive on time or not.
- It is used extensively in the field of medicine, especially during the launch of a new vaccine or drug. For instance, it helps check the probability of success of a newly-launched vaccine.
- The Bernoulli distribution can be applied to predict the weather. For e.g., to check the probability of rain.
- It is used in businesses to understand consumer behavior, buying patterns, etc.
- The concept of Bernoulli distribution also plays a crucial role in fields of machine learning and AI.
Common Mistakes and how to avoid them in the Bernoulli Distribution
Students learning about the properties, concepts, and real-life applications of the Bernoulli distribution can apply it to both their academic and real-world scenarios. Sometimes, misunderstanding the concepts of the Bernoulli distribution can lead to wrong decisions and incorrect calculations. By understanding these common errors and their helpful solutions will help students improve their problem-solving skills and analytical capabilities.
Mistake 1
Assuming the Bernoulli Distribution has more than two outcomes
Always remember that the Bernoulli distribution has only two possible outcomes. The outcomes are either success (1) or failure (0).
For e.g., when a game of tennis is played by a person, the only two possible outcomes for that person is either win or lose.
Mistake 2
Thinking that the value of p is greater than 1.
We should always keep in mind that the value of p cannot exceed 1 as the value lies between 0 and 1.
Mistake 3
Confusing the Bernoulli distribution meant for multiple trails
Students should remember that the Bernoulli distribution is for a single trial. But the Binomial distribution is used for events that have multiple trials. For e.g., tossing a coin 5 times.
Mistake 4
Misinterpreting (1 - p)
Success (p) and failure (1 - p) are the only two possible outcomes of the Bernoulli distribution. Students should remember that 1 - p represents the probability of failure, not another outcome. If the probability of success is 0.6 and the probability of failure is 0.4 (1 - p = 1 - 0.6).
Mistake 5
Misapplying the Bernoulli Distribution to continuous data
It is important to remember that the Bernoulli distribution is only for discrete outcomes.
For example, we can use it to know whether the result is head or tail (yes/no) when a coin is flipped. We cannot use this distribution for continuous outcomes. For example, the height of a person, the score on the math exam, and so on.
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Solved examples of Bernoulli Distribution
Problem 1
A factory produces light bulbs, and 3% of them are defective. If a bulb is picked randomly, what is the probability that it is defective or non-defective?
3% chance of picking a defective and a 97% chance of picking a non-defective bulb.
Explanation
Here, the formula is:
P (X = x) = px (1 - p)1 - x
(X = 1 ) = If the bulb is defective
(X = 0) = If the bulb is non-defective
Probability of success (p) = 3% or 0.03 (3 / 100 = 0.03)
The defective bulbs ( X =1):
P (X = 1) = (0.03)1 × (1 - 0.03)0
P (X = 1) = 0.03 × 1 = 0.03
For non-defective bulbs (X=0):
P (X = 0) = (0.03)0 × (1 - 0.03)1
Since (0.03)0 = 1
P (X = 0) = 1 × 0.97 = 0.97
Hence, the probability of picking a defective bulb is 0.03. While the probability of picking a non-defective bulb is 0.97.
Problem 2
A student attends school 80% of the time on any given day. What is the probability that the student attends or misses school on a randomly selected day?
80% chance of attending school and 20% chance of missing school
Explanation
Here,
Success (X = 1 ): The student attends school
Failure (X = 0 ): The student misses school
Probability of success: 80% or 0.8
Now let us apply the formula:
P (X =x) = px (1 -p)1 - x
P (X =1) = (0.8)1 × (1 - 0.8)0
P (X = 1) = 0.8 × 1 = 0.8
For missing school (X = 0):
P (X =0) = (0.8)0 × (1 - 0.8)1 = 0.2
The student has an 80%(0.8) chance of attending school and a 20% (0.2) chance of missing school.
Problem 3
A washing machine has a 70% chance of working when switched on. What is the probability that it fails?
0.3 (30%)
Explanation
The probability of success(p) = 0.7(70%)
Probability of failure (1-p) = 0.3
To find the probability of failure:
P (X = 0) = (0.7)0 × (0.3)1
1 × 0.3 = 0.3
Therefore the probability of failure is 0.3(30%).
Problem 4
A student has a 50% chance of passing an exam. What is the probability that they fail?
50%
Explanation
Success = passing the exam
Probability of success (p) = 0.5
Probability of failure (1-p) = 0.5
To find the probability of failure ( X = 0):
P (X = 0) = (0.5)0 × (0.5)1
P (X = 0) = 1 × 0.5 = 0.5
So, the probability of failing the exam is 0.5 or 50%.
Problem 5
A weather forecast says there is a 60% chance of rain today. What is the probability it won’t rain?
40% of probability.
Explanation
The probability of rain (p) = 0.6
The probability of no rain (1 -p) = 0.4
The formula for the probability of no rain is:
P (X = 0) = (0.6)0 × (0.4)1
P (X = 0) = 1 × 0.4 = 0.4
Hence, the probability of no rain is 40% or 0.4.
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FAQs on the Bernoulli Distribution
1.What do you mean by the Bernoulli distribution?
2.Explain the formula of the Bernoulli distribution.
3.What is meant by the expected value in the Bernoulli distribution?
4.Note any two characteristics of the Bernoulli distribution.
5.Differentiate Bernoulli and Binomial distribution.
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
