9. Probability and Counting

Probability is a fraction:
\[ P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}. \]
It can be written as a fraction, decimal, or percent.

Example: A bag has 5 red marbles and 3 blue marbles. There are 8 total marbles, so \(P(\text{red})=5/8\).

AND usually means multiply for independent events. The chance of flipping heads and then rolling a 6 is
\[ \frac{1}{2}\times\frac{1}{6}=\frac{1}{12}. \]

OR usually means add when the outcomes cannot both happen at the same time. Rolling a 1 or 2 on one die has probability \(\frac{1}{6}+\frac{1}{6}=\frac{1}{3}\).

Complement means "not." If rain has probability \(70\%\), no rain has probability \(30\%\).

Counting questions often use multiplication. If there are 4 shirts and 3 pants, there are \(4\times3=12\) outfits.