Introduction to Probability

What is Probability?

Probability measures how likely an event is to occur.

$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$

Probability Range

Probability is always between 0 and 1 (or 0% and 100%):

• $P = 0$: Impossible event
• $P = 0.5$: Equally likely to happen or not
• $P = 1$: Certain event

Complementary Events

The complement of event $A$ is "not $A$": $P(\text{not } A) = 1 - P(A)$

Example: If $P(\text{rain}) = 0.3$, then $P(\text{no rain}) = 1 - 0.3 = 0.7$

Independent Events

Events are independent if one doesn't affect the other.

Multiplication Rule: $P(A \text{ and } B) = P(A) \times P(B)$

Example: Flipping a coin twice: $P(\text{heads, heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Mutually Exclusive Events

Events that cannot both happen at the same time.

Addition Rule: $P(A \text{ or } B) = P(A) + P(B)$