Bernoulli Trials
Bernoulli Trials are random experiments in probability whose possible outcomes are only of two types, such as success and failure, yes and no, True and False, etc. Also, for Bernoulli trials, the probability of each outcome remains the same with each trial, i.e., each outcome is independent of the other. The process of performing Bernoulli trials is called the Bernoulli process. It was named after a Swiss mathematician, named James Bernoulli because of his significant contribution in the field of probability.
Let us understand the concept of Bernoulli trials, their condition and Bernoulli distribution along with its probability formula. We will also solve some examples for a better understanding of the concept.
1.  What are Bernoulli Trials? 
2.  Conditions of Bernoulli Trials 
3.  Bernoulli Distribution 
4.  Bernoulli Trials Formulas 
5.  FAQs on Bernoulli Trials 
What are Bernoulli Trials?
Bernoulli trials in probability are random experiments with exactly two outcomes. A reallife example of a Bernoulli trial is whether it will rain today or not. Now, the only possible outcomes are Yes and No and are independent of each other. Generally, the outcomes of a Bernoulli trial are success and failure. The probability of success is denoted by 'p' whereas the probability of failure is denoted by 1  p = q. Few other examples of Beternoulli trials are:
 If the newborn child is a girl or boy?
 The tenth card of a wellshuffled deck is an ace. The possible outcomes are Yes and No.
 The event of tossing a coin. The only two possible outcomes are heads and tails.
 Rolling a die where a '1' is a 'success', all other numbers are considered as 'failure'
Conditions of Bernoulli Trials
Now that we know the meaning of the Bernoulli trial, let us understand the conditions required for it. Given below is a list of conditions for the Bernoulli Trials:
 The number of trials should be finite.
 Each trial should be independent.
 Each trial should have only two possible outcomes  success and failure.
 The probability of each outcome should be the same in every trial.
Bernoulli Distribution
Bernoulli distribution is a discrete probability distribution of the Bernoulli random variable which takes the value 1 with probability p and the value 0 when the probability is 1 p = q. A random variable is a realvalued function whose domain is the sample space of a random experiment. This distribution has only two outcomes  success/failure, true/false, yes/no, etc. The probability of success is p and the probability of failure is 1  p = q. An example of Bernoulli distribution is cointossing where there are exactly two possible outcomes  Heads and Tails.
Bernoulli Trials Formulas
Some of the important formulas related to the Bernoulli Trials are given below:
 The probability of x if x is a random variable in Bernoulli distribution
P(x = 1) = p, P(x = 0) = 1  p = q  If X is the number of successes in a Binomial experiment with n independent trials, then
P(X = k) = ^{n}C_{k} p^{k }q^{nk}, where p is the probability of success and q is the probability of failure.  The probability mass function PMF for Bernoulli distribution (when n = 1 in binomial distribution) where z is a random variable and p is the probability of succes is
f(z, p) = {p, if z = 1 and q = 1 p, if z = 0}
OR
f(z, p) = p^{z }(1  p)^{1z}, for z = 0, 1
OR
f(z, p) = pz + (1  p)(1  z), for z = 0, 1  The mean (expected value) of a Bernoulli random variable X is
E(X) = p  The variance of Bernoulli random variable X is
Var[X] = p(1  p) = pq
Important Notes on Bernoulli Trials
 Bernoulli trials have only two possible outcomes.
 The two possible outcomes are independent of each other.
 The probability of success is p and the probability of failure is 1  p = q.
 The probability of each outcome in each Bernoulli trial remains the same.
Related Topics on Bernoulli Trials
Bernoulli Trials Examples

Example 1: James draws balls 5 times from a bag of 10 balls which includes 5 red and 5 green balls with replacement. Check if this is an example of the Bernoulli trials.
Solution: We need to check if all conditions of the Bernoulli trials are satisfied.
James draws the ball 5 times, that is, the number of trials is 5 which is finite.
Since the ball is replaced, each trial is independent.
There are only two possible outcomes  red and green
The probability of drawing a red ball = probability of drawing a green ball = 5/10 = 1/2
This implies all conditions of the Bernoulli trials are satisfied.
Answer: The given example is a Bernoulli experiment.

Example 2: A football player 7 independent free shots with a probability of 0.6 of getting a goal on each shot. Determine the number of trials and the probability of not getting a goal in each shot.
Solution: The given case is an example of Bernoulli trials as the number of trials is 7 which is finite.
There are only two possible outcomes  getting a goal and not getting a goal.
Each trial is independent of the other and the probability of getting a goal = 0.6 = Probability of success
⇒ Probability of not getting a goal = Probability of failure = 1  0.6 = 0.4
Answer: The number of trials is 7 and the probability of not getting a goal in each shot is 0.4.
FAQs on Bernoulli Trials
What are Bernoulli Trials in Probability?
Bernoulli Trials are random experiments in probability whose possible outcomes are only of two types, such as success and failure, yes and no, True and False, etc. Also, for Bernoulli trials, the probability of each outcome remains the same with each trial, i.e., each outcome is independent of the other.
How many Trials are there in Bernoulli Trials?
The number of trials in Bernoulli trials is finite.
What is the Relationship Between Bernoulli Distribution and Binomial Distribution?
Multiple Bernoulli trials make up the Binomial experiment. So, when the value of n (number of trials) is 1 in a binomial distribution, it is called a Bernoulli distribution.
How to Tell Bernoulli Trials?
If the trials satisfy the below conditions, then they are called Bernoulli trials:
 The number of trials should be finite.
 Each trial should be independent.
 Each trial should have only two possible outcomes  success and failure.
 The probability of each outcome should be the same in every trial.
Is Rolling a Die a Bernoulli Trial?
Rolling a die is a Bernoulli Trial only if one number of the six outcomes are clubbed into two possible outcomes only as success and failure. For example, when rolling a die, getting an even number is a success whereas getting an odd number is a failure. Another example is getting 2 is considered as a success, else it is a failure. If all six outcomes are considered individually, then it cannot be considered as a Bernoulli trial as it exceeds the number of possible trials for Bernoulli trials.
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