Overview

Intermediate probability uses conditioning and structure to simplify counting. Expectations, states, and geometric probability appear frequently at this level, especially on AMC 12 and AIME.

Key Ideas

• $P(A\mid B)=P(A\cap B)/P(B)$.
• Expected value is linear: $E[X+Y]=E[X]+E[Y]$.
• Use complementary events to avoid messy direct counts.
• Independence check: $P(A\cap B)=P(A)P(B)$.
• For repeated trials, define states and write equations.

Core Skills

Condition on a Step

Break a problem into cases based on the first draw/roll, then use total probability.

Use Linearity of Expectation

Compute expected values by summing indicators, even without independence.

Build a State Equation

For multi-stage processes, define states and set up equations for $P_i$.

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 $\frac{4}{52}\cdot\frac{3}{51}=\frac{1}{221}.$

Conditional Probability in Practice

Two dice are rolled. Given the sum is at least 9, find the probability both dice are at least 4.

Outcomes with sum at least 9: 10 outcomes. Outcomes with both at least 4 and sum at least 9: 8 outcomes. So the answer is $8/10=4/5$.

Expected Value

Linearity of expectation works even without independence. Example: expected number of aces in 5 cards is $5\cdot\frac{4}{52} = 5/13$.

Geometric Probability

Choose a random point in the unit square. The probability that $x+y\lt 1$ equals the area of a right triangle with area $1/2$, so the answer is $1/2$.

States Method (Probability)

For a process that moves between states, let $P_i$ be the probability of success starting in state $i$. Then

$P_i = \sum_j p_{ij} P_j$

with boundary values on terminal states.

Example: A frog starts at 1 on the line $\{0,1,2,3,4\}$, moves right with probability $2/3$ and left with probability $1/3$. If it reaches 0 it fails; if it reaches 4 it succeeds. Solve the linear system to get $P_1=8/15$.

Strategy Checklist

• Define the sample space or state diagram first.
• Condition on a natural first step.
• Use complements for "at least" events.
• Check independence before multiplying.

Common Pitfalls

• Treating dependent events as independent.
• Forgetting to update counts after conditioning.
• Omitting boundary states in state equations.

Practice Problems