- Understand the fundamental concepts of probability.
- Learn about probability rules and axioms.
- Explore conditional probability and Bayes’ Theorem.
- Visualize probability distributions.
Definition of Probability
Probability measures the likelihood of an event occurring. It is defined as:
\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]Basic Rules of Probability
| Rule | Formula | Description |
|---|---|---|
| Complement Rule | \( P(A^c) = 1 - P(A) \) | The probability of an event not occurring. |
| Addition Rule (for mutually exclusive events) | \( P(A \cup B) = P(A) + P(B) \) | If events A and B cannot happen together. |
| Multiplication Rule (for independent events) | \( P(A \cap B) = P(A) P(B) \) | If A’s occurrence does not affect B. |
Conditional Probability and Bayes’ Theorem
Conditional probability is:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]Bayes’ Theorem:
\[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} \]Proof
Starting from conditional probability:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]Similarly:
\[ P(B | A) = \frac{P(A \cap B)}{P(A)} \]Rearrange:
\[ P(A \cap B) = P(B | A) P(A) \]Substituting in:
\[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} \]Visualizing Probability Distribution
Exercises
- Question 1: A bag contains 4 red and 6 blue balls. What is the probability of drawing a red ball?
- Question 2: If \( P(A) = 0.4 \), \( P(B) = 0.5 \), and \( P(A \cap B) = 0.2 \), find \( P(A \cup B) \).
- Question 3: What is the probability of flipping two fair coins and getting at least one heads?
- Answer 1: \( \frac{4}{10} = 0.4 \)
- Answer 2: \( P(A \cup B) = 0.7 \)
- Answer 3: \( \frac{3}{4} \)