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Probability Theory

  • 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: \( \frac{3}{4} \)

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