Bernoulli Trials

Bernoulli trials are repeated, independent experiments in which each trial has only two possible outcomes, usually labeled 1 (success) and 0 (failure). For example, will it rain today? The possible outcomes are yes or no, and these outcomes are independent.

The probability of success (p) remains the same for every trial, no matter how many times you repeat the experiment. Then the probability of failure is \(q = 1 - p\). 

For example, consider tossing a fair coin once. 

Here, getting heads may be called a success (1)

Getting tails may be called a failure (0)

The probability model of a single Bernoulli trial is described using the Bernoulli distribution. A Bernoulli distribution is the probability distribution of a random variable that takes only two values:
 

• 1 (success) with probability p
   
• 0 (failure) with probability 1 - p

For example, if the probability that the tenth card drawn from a shuffled deck is an ace is \(p = {1\over 13}\). 

Define a random variable X as: 

• X = 1 if the tenth card is an ace
   
• X = 0 otherwise
   

Then X follows a Bernoulli distribution with parameter \(p = {1\over 13}\)

Therefore: \(P(X =1) = {1\over13} \\ \ \\ P(X = 0) = {12\over 13 } \)

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 

Bernoulli trials are the foundation of many probability experiments. Mastering them helps you understand independent events and predict outcomes with confidence.

• Understand that each trial has only two possible outcomes, success and failure.
   
• Learn the relationship between probability of success (p) and failure (1-p).
   
• Practice identifying Bernoulli trials in real-life examples like coin tosses or product testing.
   
• Use formulas for mean and variance to analyze repeated Bernoulli experiments.
   
• Solve multiple practice problems to strengthen your understanding of independent trials.
   
• Parents can encourage hands-on experimentation by letting students perform simple activities such as tossing a coin, rolling a die, or picking colored balls. These activities make learning interactive and fun.
   
• Teachers can help students understand Bernoulli trials by using simple yes-or-no situations from daily life to illustrate how each trial has two possible outcomes.
   

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.
   
• Manufacturing: Bernoulli trials are used to check product quality. Each item inspected either passes or fails the quality test, representing a success or failure outcome in a single trial.