Counting Practice this

Order matters → permutations: ⁿP_k = n!/(n−k)!. Order doesn't → combinations: C(n,k) = n!/(k!(n−k)!). Divide out repeated items.

Complements Practice this

"At least one" is almost always 1 − P(none).

Independence Practice this

A, B independent ⇔ P(A and B) = P(A)·P(B). Draws without replacement are not independent.

Conditional probability Practice this

P(A | B) = P(A ∩ B) / P(B): shrink the sample space to B, then count.

Bayes' rule Practice this

P(H | E) = P(E | H)·P(H) / P(E). Faster in odds: posterior odds = prior odds × likelihood ratio. Or count 1,000 imaginary people.

Law of total probability Practice this

P(A) = Σ P(A | Bᵢ)·P(Bᵢ) over a partition. In games, condition on the first step and solve for the unknown.

Binomial Practice this

k successes in n independent trials: C(n,k)·p^k(1−p)^n−k. Mean np, variance np(1−p).

Geometric Practice this

Trials until the first success: P(N = k) = (1−p)^k−1p, E[N] = 1/p, P(N > k) = (1−p)^k.

Symmetry Practice this

If nothing distinguishes the outcomes, they are equally likely. Look for it before computing anything.

Linearity of expectation Practice this

E[X + Y] = E[X] + E[Y], always, even when dependent. Write a count as a sum of 0/1 indicators.

Variance basics

Var(X) = E[X²] − E[X]². Var(aX + b) = a²Var(X). For independent X, Y: Var(X + Y) = Var(X) + Var(Y). SD scales with √n.

EV of a bet Practice this

EV = Σ payoff × probability. Include the losses as negative payoffs.

Fair price Practice this

A game's fair price is its expected payoff. Paying less is +EV; paying more is −EV.

Optimal stopping Practice this

Work backwards. At each step, stop if what you hold beats the expected value of continuing.

Risk vs EV Practice this

EV ignores variance and ruin. Insurance is −EV for the buyer but can still be sensible when the loss would hurt. Capped payouts tame "infinite" EV.

Kelly sizing (intro)

For a bet paying b-to-1 with win probability p (q = 1 − p), the growth-optimal stake is f* = p − q/b of bankroll. Betting twice Kelly gives roughly zero long-run growth; many traders use half-Kelly.

From EV to a two-sided market Practice this

1. Estimate fair value (FV). 2. Quote a bid below and an offer above it. 3. Make it wider when you are unsure or the payoff is volatile. 4. Update after each trade: if you're lifted (they buy your offer) you are now short and they may know more, so move both prices up. If you're hit (they sell to you at your bid), move down.

Sum of two dice Practice this

Ways to make s: 6 − |s − 7|. Mean 7, variance 35/6.

Reciprocals Practice this

Card probabilities Practice this

Powers of 2

Squares to 30² Practice this

Dice order statistics Practice this

×5 Practice this

Multiply by 10, halve.

×25 Practice this

Multiply by 100, divide by 4.

×125 via division Practice this

125 = 1000/8: multiply by 1000, halve three times.

×11 Practice this

Two digits: write the digits apart and put their sum between them, carrying if needed.

Near 100 (base method) Practice this

Write each as 100 ± d. Cross-add for the hundreds; multiply the d's for the rest.

Squaring numbers ending in 5 Practice this

Tens digit t: t·(t+1), then append 25.

Difference of squares Practice this

(a + b)(a − b) = a² − b². Spot pairs equidistant from a round number.

Percentage swap Practice this

x% of y = y% of x. Pick the easier direction.

Split the multiplication Practice this

Break one factor into place values.

÷8 and ÷16 by halving Practice this

Halve three (or four) times.

Compensation Practice this

Round one number, then fix the difference.

Estimation and sanity checks Practice this

Round to two significant figures, then correct by the rounding percentage. Always check the order of magnitude.