AMC 10/12 · Cheatsheet

Topic 3 · Counting & Probability

Chapter 3 · Probability

📋 Reference · always available

Basic

P = \tfrac{\#\text{favorable}}{\#\text{total}}

Independent

P(A\text{ and }B) = P(A)P(B)

Complement

P(\ge1) = 1 - P(\text{none})

Geometric

Probability = favorable area ÷ total area.

Conditional

P(A|B) = \tfrac{P(A\cap B)}{P(B)}

Bayes

P(A|B) = \tfrac{P(B|A)\,P(A)}{P(B)}

Total probability

P(B) = P(B|A)P(A) + P(B|\neg A)P(\neg A)

Markov recursion

p_k = \sum_\ell P(\text{transition }k\!\to\!\ell)\,p_\ell

with boundary conditions; solve linearly or telescope.

Weighted mean

\bar x = \tfrac{\sum w_i x_i}{\sum w_i}

Median

2k{+}1

values: middle =

(k{+}1)^{\text{th}}

2k

values: avg of

k^{\text{th}}, (k{+}1)^{\text{th}}

Unique mode

Strict inequality: every other value appears strictly fewer times.