Probability
The language of uncertainty. Probability measures the likelihood of an event occurring.
Basic Concepts
- Experiment: A process with uncertain outcomes (e.g., rolling a die).
- Sample Space ($S$): Set of all possible outcomes (e.g., {1, 2, 3, 4, 5, 6}).
- Event ($A$): A subset of outcomes (e.g., Rolling an even number {2, 4, 6}).
Rules of Probability
1. Addition Rule (Union)
Probability of A OR B.
$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$
If mutually exclusive, $P(A \cap B) = 0$.
2. Multiplication Rule (Intersection)
Probability of A AND B.
$$ P(A \cap B) = P(A) \times P(B|A) $$
If independent, $P(B|A) = P(B)$, so $P(A \cap B) = P(A) \times P(B)$.
Bayes' Theorem
Updating probabilities based on new information.
$$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$
Medical Test
A test is 99% accurate. Disease is rare (1%). If you test positive, what is the chance you actually have the disease?
It's often much lower than 99% due to false positives!
Test Yourself
Q1: If two events cannot happen at the same time, they are: