Different Approaches to Probability Theory
Data Science and AI Lecture Series
Author: Bindeshwar Singh Kushwaha
Introduction
- Classical probability has limitations when outcomes are not equally likely or finite.
- Alternative approaches are needed in situations where classical definitions fail.
- This unit introduces methods based on past experiences, observed data, and axioms.
- Topics discussed include:
- Relative frequency approach (statistical probability)
- Subjective probability
- Axiomatic probability
Relative Frequency Approach and Statistical Probability
Classical probability fails when:
- Outcomes are not equally likely.
- The number of exhaustive cases is infinite.
Relative frequency approach observes data to compute probabilities.
Statistical (or empirical) probability is defined as:
\[ P(A) = \lim_{n \to \infty} \frac{m}{n} \]
Where:
- m: Number of times event A occurs.
- n: Total number of trials under identical conditions.
Example 1: Salary Distribution
Example: Probability that an employee’s salary is less than Rs. 150.
| Salary Range (Rs.) | Number of Employees |
|---|---|
| Below 100 | 20 |
| 100-150 | 40 |
| 150-200 | 50 |
| 200 and above | 15 |
Total employees: \( 20 + 40 + 50 + 15 = 125 \)
Employees with salary \(< 150\): \( 20 + 40 = 60 \)
Probability:
\[ P(\text{Salary} < 150) = \frac{60}{125} = 0.48 \]
Example 2: Coin Toss Experiment
Toss a coin 200 times and record the number of heads. The data is shown below:
| Number of Tosses (n) | Number of Heads (m) | Proportion \( \frac{m}{n} \) |
|---|---|---|
| 1 | 1 | 1.0 |
| 2 | 2 | 1.0 |
| 3 | 2 | 0.67 |
| 4 | 3 | 0.75 |
| 10 | 6 | 0.6 |
| 50 | 29 | 0.58 |
| 200 | 105 | 0.525 |
Observation:
\[ \lim_{n \to \infty} \frac{m}{n} = 0.5 \]
Probability of getting heads is \( \frac{1}{2} \).
Limitations of Statistical Probability
- Experimental conditions may change over time.
- \( \lim_{n \to \infty} \frac{m}{n} \) may not converge to a unique value.
Video
Reach PostNetwork Academy
- Website: www.postnetwork.co
- YouTube Channel: www.youtube.com/@postnetworkacademy
- Facebook Page: www.facebook.com/postnetworkacademy
- LinkedIn Page: www.linkedin.com/company/postnetworkacademy
Thank You!