Bernoulli Trials and Binomial Distribution

Bernoulli Trials

Bernoulli trial, binomial distribution and Bernoulli distribution are briefly explained in this article.Let us first learn about Bernoulli trials.

Bernoulli trials are also known as binomial trials as there are only possible outcomes in Bernoulli trials i.e success and failure whereas in a binomial distribution, we get a number of successes in a series of independent experiments. A Bernoulli distribution is the probability distribution for a series of Bernoulli trials where there are only two possible outcomes. In this article, we will discuss,bernoulli trial binomial distribution, bernoulli trial formula, bernoulli trial example, bernoulli distribution, bernoulli distribution examples, properties of bernoulli distribution, how bernoulli trial is related to binomial distribution etc. 

Binomial Distribution

The binomial distribution is a kind of probability distribution that has two possible outcomes. In probability theory, binomial distributions come with two parameters such as n and p.

The probability distribution becomes a binomial probability distribution when it satisfies the below criteria.

  1. The number of trials must be fixed.

  2. The trials are independent of each other.

  3.  The success of probability remains similar for every trial.

  4. Each trial has only two outcomes namely success or failure.

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Bernoulli trial

A random experiment that has only two mutually exclusive outcomes such as “success “and “not success “is known as a Dichotomous experiment.

If a Dichotomous experiment is repeated many times and if in each trial you find the probability of success p (0< p <1) is constant, then all such trials are known Bernoulli trials.

As, Bernoulli trials has only two possible outcomes, it can easily frame as “yes” or “no” questions

Bernoulli Trials Example

The Bernoulli trial example will explain the concept of bernoulli trial in two different situation:

8 balls are drawn randomly including 10 white balls and 10 black balls. Examine whether the trials are Bernoulli trials if the balls are replaced and not replaced.

  1. In the first trial, when the ball is drawn with replacement, the probability of success (say, the black ball) is 10/20 = ½ which is similar for all 8 trials. Hence, the trials including the drawing of balls with replacement are considered as Bernoulli trials.

  1. In the second trial, when drawn without replacement, the probability of success (say, the black balls) changes with the number of trials =10/20 = ½ for second trials, the probability of success p =9/19 which is not similar to the first trial. Hence, the trails including the drawing of balls without replacement are not considered as Bernoulli trial

Bernoulli Trials Conditions

  • The Bernoulli trial has only two possible outcomes i.e. success or failure.

  • The probability of success and failure remain the same throughout the trials.

  • The bernoulli trials are independent of each other.

  • The number of trials is fixed.

  • If x is the probability of success then probability of failure is 1-x.

Bernoulli trials Formula

Here, you can see the Bernoulli trial formula in Bernoulli Maths.

Let us take an example where n bernoulli trials are made then the probability of getting r successes in n trials can be derived by the below- given bernoulli trials formula.

P(r) = Cn pr qn-r

The term n! / r!(n!-r!)!  is known as a binomial coefficient.

Bernoulli Trials and Binomial Distribution

  • The student will be able to design a  Bernoulli trial or experiment

  • The student will be easily able to use binomial formula

  • The student will be able to design a  binomial distributions

  • The students will be able to compute applications including Bernoulli trials and binomial distribution’

Bernoulli Distribution

A bernoulli distribution in bernoulli Maths is the probability distribution for a series of bernoulli trials where there are only two possible outcomes. It is a kind of discrete probability distribution where only specific values are possible. In such a case, only two values are possible;e ( n=0 for failure and n=1 for success). This makes the Bernoulli distribution the simplest form of the probability distribution that persists.

Bernoulli Distribution Examples

Some of the bernoulli distribution examples given in bernoulli Maths are stated below:

  1. A newly born child is either a girl or a boy ( Here, the probability of a child being a  boy is roughly 0.5)

  2. The student is either pass or fail in an exam

  3. A tennis player either wins or losses a match

  4. Flipping of a coin is either a head or a tail.

Properties of Bernoulli distribution

Here, you can find some of the properties of bernoulli distribution in bernoulli Maths.

  1. The expected value of the bernoulli distribution is given below.

E(X) = 0 * (1-P) + 1 * p = p

  1. The variance of the bernoulli distribution is computed as 

Var (X) = E(X²) -E(X²) = 1² * p +0² * ( 1-p) - p² = p - p² = p (1-p)

  1. The mode, the value with the highest probability of appearing, of a Bernoulli distribution is 1 if p > 0.5 and 0 if < 0.5, success and failure are equally likely and both 0 and 1 are considered as modes. 

  2. The basic properties of bernoulli distribution can be computed by considering n=1 in probability

Quiz Time

  1. How many possible outcomes can be there for  bernoulli trials?

  1. More than 1

  2. More than 2

  3. Exactly 1

  4. Exactly 2

      2.  What will be the variance of the Bernoulli trials, if the probability of success of the Bernoulli trial is  0.3.

  1.  0.3

  2. 0.7

  3. 0.21

  4. 0.09

      3. The mean and variance are equal in binomial distributions.

  1. True

  2. False

Solved Examples

  1. If the probability of the bulb being defective is 0.8, then find the probability of the bulb not being defective.


Probability of bulb being faulty, p = 0.8

Probability of bulb not being defective, q = 1-p = 1-0.8= 0.2

 Hence, probability of bulb not being defective, q = 0.2


In an exam, 10 multiple choice questions are asked where only one out of four questions are  correct. Find the probability of getting 5 out of 10 questions correct in an answer sheet.

Solution: Probability of getting an answer correct, p = ¼

Probability of getting an answer incorrect , q = 1-p = 1

Probability of getting 5 answers correct, P(X=5) = (0.25)5 ( 0.75)5 = 0.5839920044

3. Toss a coin for 12 times. What is the probability of getting 7 heads? 


Number of trials(N) = 12

Number of success (r)= 7

Probability of single trial  (P) = ½ = 0.5

nCr =

[n!r!] * (n-r)!

12!7! (12-7)!

= 95040120

= 792

pr= 0.57=


To find (1-p)n-r,calculate (1-p) and (n-r)

Solving P (X=r) = nCr .pr. (1-p)n-r

= 792 * 0.0078125 *0.03125

= 0.193359375

The probability of getting 7 heads is 0.19

FAQ (Frequently Asked Questions)

1. How is the Bernoulli trial related to the binomial distribution?

Both are types of the discrete probability distribution that obtains the probability of success in an outcome. The outcomes in each distribution are independent.

Bernoulli distribution is a distribution with only two possible outcomes; “yes” with probability p and “no” with probability 1-p. For example, tossing a coin has two possible outcomes. Head which can be referred to as “yes” or Trial which can be referred to as “no”. The Bernoulli distribution is used only for a single trial. If multiple repeated n Bernoulli trials are carried out with p probability of success, the distribution becomes binomial. For example, tossing a coin five times is a binomial experiment.

2. What is the important part of Bernoulli trial?

The important part of Bernoulli trial is that every action must be independent. It implies that probability must remain similar throughout the trials; each event must be completely separate and have nothing to do with the previous event.

Winning a scratch- off lottery is considered an independent event. Your chances of winning each ticket are the same as winning on another ticket. On the other hand, drawing lotto numbers is considered an independent event. Lotto numbers come of a ball (the numbers are not replaced) so the probability of successive numbers being picked relies on the number of balls left, when there are fifty balls, the probability is 1/50 that any numbers of balls are picked but when there are only 5 balls are left, the probability shoots up 1/5. Although it is possible to find those probabilities but it is not a Bernoulli trial because the events (picking the numbers) are related to each other.