1. Introduction to Probability Distributions
A probability distribution is a mathematical function that describes the possible outcomes of a random variable and the probability of each outcome. Three commonly used probability distributions in business, economics, engineering, and healthcare are:
- Binomial Distribution – Models the probability of a fixed number of successes in a given number of trials.
- Poisson Distribution – Models the probability of a certain number of events occurring in a fixed time period.
- Normal Distribution – Models the probability of data being distributed symmetrically around a mean value.
Each of these distributions is used in real-world decision-making, forecasting, and risk management.
2. Binomial Distribution
(i) Concept of Binomial Distribution
The Binomial Distribution describes the probability of obtaining a specific number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure).
A binomial experiment must meet the following conditions:
✔ Fixed number of trials (nnn).
✔ Each trial has two outcomes: Success (ppp) or Failure (1−p1 – p1−p).
✔ Trials are independent (outcome of one trial does not affect another).
✔ Probability of success remains constant.
Formula for Binomial Probability
Where:
- n = Total number of trials
- k = Number of successes
- p = Probability of success
- (nk) = Combination formula: n!/ k!(n−k)!
(ii) Application of Binomial Distribution
Example 1: Quality Control in Manufacturing
A factory produces LED bulbs, and 95% of them are non-defective. If a random sample of 10 bulbs is tested, what is the probability that exactly 9 bulbs are non-defective?
Given:
- n=10 (Total bulbs tested).
- k=9 (Number of non-defective bulbs).
- p=0.95 (Probability of non-defective bulb).
Using the formula:
So, the probability is 31.5%.
✔ Used in product defect rate analysis.
Example 2: Marketing Campaign Success
A company runs an email marketing campaign where the probability of a customer clicking on an ad is 20%. If 15 customers receive the email, what is the probability that exactly 3 customers click on the ad?
Given:
So, the probability is 25.08%.
✔ Used in digital marketing performance measurement.
3. Poisson Distribution
(i) Concept of Poisson Distribution
The Poisson Distribution is used to model the probability of a certain number of events occurring within a fixed time period or space, where events occur randomly and independently.
A Poisson experiment must meet the following conditions:
✔ Events occur one at a time.
✔ Events occur at a constant average rate (λ).
✔ Each event is independent of the previous one.
Formula for Poisson Probability
Where:
- λ = Expected number of events in a time period.
- k = Exact number of occurrences.
- e = Euler’s number (~2.718).
(ii) Application of Poisson Distribution
Example 1: Customer Service Call Rates
A call center receives an average of 6 calls per hour. What is the probability that exactly 4 calls arrive in the next hour?
Given:
- λ = 6 calls per hour.
- k = 4
So, the probability is 13.38%.
✔ Used in workforce planning for call centers.
Example 2: Traffic Flow Analysis
A traffic light sees 10 cars per minute on average. What is the probability that exactly 8 cars pass in a given minute?
Given:
- λ=10.
- k=8.
Using calculations, P(8)≈0.1126 (11.26%).
✔ Used in traffic engineering and city planning.
4. Normal Distribution
(i) Concept of Normal Distribution
The Normal Distribution (also called the Gaussian Distribution) describes a probability distribution where data is symmetrically distributed around the mean. It follows a bell-shaped curve.
A Normal Distribution must meet these conditions:
✔ Mean (μ\muμ) and standard deviation (σ\sigmaσ) define the shape.
✔ Most data points are near the mean.
✔ Follows the 68-95-99.7 rule (empirical rule).
Formula for Normal Probability Density Function
Where:
- μ = Mean.
- σ = Standard deviation.
- e = Euler’s number (~2.718).
(ii) Application of Normal Distribution
Example 1: Employee Performance Evaluation
If the average employee performance score is 70 with a standard deviation of 10, what percentage of employees score between 60 and 80?
Using the Z-score formula:
From the standard normal table, the area between Z = -1 and Z = 1 is 68.26%.
✔ Used in employee performance measurement and hiring tests.
Example 2: Stock Market Returns
Stock market returns are normally distributed. If the average return of a stock is 8% per year with a standard deviation of 5%, what is the probability of a return greater than 15%?
Using the Z-score:
From the normal table, the probability of exceeding this is 8.08%.
✔ Used in financial risk assessment.
5. Conclusion
- Binomial Distribution → Used for success/failure experiments (e.g., quality control, marketing).
- Poisson Distribution → Used for counting events over time (e.g., call rates, traffic).
- Normal Distribution → Used for data that follows a bell curve (e.g., stock returns, employee scores).