Overview

Intermediate probability uses conditioning and structure to simplify counting.

Key Ideas

• $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$.
• Expected value is linear: $E[X+Y]=E[X]+E[Y]$.
• Use complementary events to avoid messy direct counts.

Worked Example

Two cards are drawn without replacement from a standard deck. Find the probability both are aces.

There are $4$ aces in $52$ cards. The probability is $\frac{4}{52}\cdot\frac{3}{51}=\frac{1}{221}.$

Practice Problems