Probability Basics

What Is Probability?

Probability measures how likely an event is to occur, on a scale from 0 to 1:

$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Probability Scale

| Value | Meaning | |-------|---------| | $0$ | Impossible | | $0.25$ | Unlikely | | $0.5$ | Equally likely | | $0.75$ | Likely | | $1$ | Certain |

Theoretical vs. Experimental Probability

Theoretical: Based on reasoning about equally likely outcomes. $P(\text{heads}) = \frac{1}{2}$

Experimental: Based on data from actual experiments. $P(\text{heads}) = \frac{\text{number of heads}}{\text{total flips}}$

As the number of trials increases, experimental probability gets closer to theoretical probability. This is the Law of Large Numbers.

Compound Events

Independent events: One event doesn't affect the other. $P(A \text{ and } B) = P(A) \times P(B)$

Example: Flip a coin AND roll a die: $P(\text{heads and 6}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$

"Or" Probability

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Sample Spaces

List all possible outcomes using:

• Tree diagrams
• Tables
• Organized lists

Complement

$P(\text{not } A) = 1 - P(A)$

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

Key idea: Probability is always between 0 and 1. If you get a value outside this range, check your work!