Bernoulli Trials

Bernoulli trials are the independent trials in which the probability of getting a possible outcome remains the same. In probability experiments, no matter how many times the same experiment is conducted, the outcome will be either 1 (success) or 0 (failure).

The concept was introduced by a Swiss mathematician, Jacob Bernoulli, in the 17th century.

For example, whether Alan wins a running match.

Here, the two possible results or outcomes are success or failure, and these answers are independent of each other.    

Bernoulli trials are important in probability and many probability theories are established based on these trials. However, the Bernoulli trials need to meet certain conditions. They are listed below: 

• The Bernoulli trials are independent.
   
• Success or failure are the two possible outcomes of a Bernoulli trial. 
   
• In every trial, the probability of getting the same outcome remains the same. 
   

Bernoulli trials help us predict the probability of an event happening or not. In our daily lives, the formulas related to the Bernoulli trials assist in calculating the probabilities and estimating the outcomes in advance.
 

1. We can calculate the probability of success or failure in a probability distribution by using the formula:

             P(x = 1) = p,  
             P(x = 0) = 1 - p = q

Here, x is a random variable in a Bernoulli distribution. 

p is the probability of success. 

q = 1 - p is the probability of failure.

2. In a binomial experiment that has a number of independent Bernoulli trials, the number of successes is represented by X. Then the formula is:

            P (X = k) = ⁿC_k p^kq^n-k 

Here, k is the desired number of successes. 

p is the probability of success in each trial.

^`C_k represents that n chooses k.   

q is the probability of failure. 

3. When we calculate the probability mass function (PMF) of the Bernoulli distribution when n = 1 in the binomial distribution, the z is a random variable and p is the probability of success:  

\(f(n) = \begin{cases} p & \quad \text{if } z=1,\\ q=1-p & \quad \text{if } z=0. \end{cases} \)

Or,  

f(z, p) = p^{​​​​​​z} (1 - p)^{​​​​​​1-z}, for z = 0, 1    

Or,

f(z, p) = p^{​​​​​​z} × (1 - p)^{​​​​​1-z}, for z = 0, 1
 

4. A Bernoulli random variable X’s mean or the expected value can be calculated by using the formula: 

          E(X) = p


5. A Bernoulli random variable X’s variance can be calculated as: 

         Var[X] = p(1 - p) = pq 

In our daily lives, we have lots of situations that only have two possible results. Bernoulli trials are widely used in various fields and scenarios to calculate probabilities.  
 

• Medical field: To check the effectiveness of a new drug or a patient’s recovery status, medical professionals use the Bernoulli trials.
   
• Sports: Coaches, and analysts, can use the Bernoulli trials to calculate the chances of winning and failing.  
   
• Marketing: The marketing and research teams can analyze the customer’s behavior by using the Bernoulli trials. If a company introduces a new online advertisement and no one clicks on it, that means failure in terms of engagement. 
   
• Education/Elections: To predict the performance of students and analyze the results of an election, Bernoulli trials can be used by teachers and pollsters.