Overview

Probability questions often reduce to counting. Define a clear sample space, count favorable outcomes, and use complements when easier.

Key Ideas

• $P(A) = \frac{\text{favorable}}{\text{total}}$ when outcomes are equally likely.
• $P(A^c) = 1 - P(A)$ is often simpler to compute.
• Use the rule of product to count outcomes in multi-step experiments.

Worked Example

Two fair dice are rolled. What is the probability the sum is $7$?

There are $36$ equally likely outcomes. The favorable pairs are $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$, so the probability is $\frac{6}{36}=\frac{1}{6}$.

Common Pitfalls

• Assuming outcomes are equally likely without checking.
• Forgetting to count order when it matters.

Practice Problems