Methods Library
Each method page answers one question: when should this technique come to mind?
Bayes' Theorem
Reverse a conditional probability: go from P(evidence | cause) to P(cause | evidence).
33 practice questions
Stars & Bars
Count ways to split identical items among labeled groups.
15 practice questions
Indicator Variables
Write a count as a sum of 0/1 variables, then add their probabilities.
18 practice questions
Linearity of Expectation
The expectation of a sum is the sum of the expectations — always.
37 practice questions
Complement Rule
Compute the probability that something never happens, then subtract from 1.
17 practice questions
Symmetry
When outcomes are interchangeable, their probabilities are equal.
17 practice questions
Inclusion-Exclusion
Count a union by adding singles, subtracting pairs, adding triples…
13 practice questions
Geometric Probability
Turn continuous uniform probability questions into areas.
29 practice questions
Order Statistics
Work with the sorted values: minimum, maximum and the $k$-th smallest.
38 practice questions
Tail-Sum Formula
For non-negative variables, the mean is the sum (or integral) of tail probabilities.
8 practice questions
Conditional Expectation
Average over a hidden quantity: $E[X] = E[E[X \mid Y]]$.
12 practice questions
Recursive Expectation / First-Step Analysis
Condition on the first move and write equations linking the states.
20 practice questions
Memorylessness
For exponential and geometric waits, the past does not change the future.
41 practice questions
Dynamic Programming / Continuation Value
Solve decision problems backwards: act when the current value beats the continuation value.
17 practice questions
Gap Method
Place the unrestricted items first, then put the restricted ones in the gaps.
15 practice questions
Circular Block Method
Fix one seat to remove rotations; glue items that must sit together.
9 practice questions
Multinomial Counting
Arrange items with repeated types: divide $n!$ by the repeats.
8 practice questions
Conditioning on the First Event
Split on what happens first, then notice the game often restarts.
2 practice questions
Exponential Race
Among independent exponential clocks, each rings first with probability proportional to its rate.
26 practice questions
Poisson Thinning
Randomly keep each Poisson event with probability $p$: you get a Poisson process of rate $\lambda p$.
14 practice questions
No-Arbitrage Replication
Price a payoff by building a portfolio that copies it; equal payoffs must have equal prices.
18 practice questions
Lagrange Multipliers
Optimize with a constraint by matching the gradient of the objective to the constraint's gradient.
20 practice questions
Jensen's Inequality
For convex $f$, $E[f(X)] \ge f(E[X])$.
6 practice questions
Change Variables / Normalize
Rescale or transform the problem into its simplest equivalent form.
1 practice questions
Pigeonhole Principle
More items than boxes forces at least one box to hold two.
4 practice questions