The Probability House

Every probability question is a room in the same house. Fill the house once from what you are given — then each formula is just a different room you read off.

One house, many readings. A and A′ down the side, B and B′ across the top. The four inner cells are the joint chances; they always sum to 1. The margins are the totals P(A), P(B). Union reads the margins minus the overlap. Conditional shrinks the world to one column. Total probability sums the two lanes into a margin. Bayes flips a forward chance into its reverse. Same house — you just stand in a different room.

The House

One contingency table

The cells the active path touches light up in purple. Given values are blue; everything else is derived.

	B	B′	Σ rad
A
A′
Σ kol

given this path derived

Connected Spells

The Probability House builds the mass. The Weighing Table takes those probabilities and measures the spread. Drainage counts the ways to fill a cell when you cannot read it directly.

Formula Reference — Kap 2 + 4

Addisjon — UnionKap 2

P(A∪B) = P(A) + P(B) − P(A∩B)

At least one of A or B. Add the two margins, then subtract the overlap you counted twice.

Multiplikasjon — JointKap 4

P(A∩B) = P(A|B) · P(B) = P(B|A) · P(A)

Both at once. A conditional times the margin it was conditioned on. Read either way round.

Betinget — ConditionalKap 4

P(A|B) = P(A∩B) / P(B)

Shrink the world to the B-column, then ask what share of it is also A.

Oppsplitting — Total ProbabilityKap 4

P(B) = P(B|A)·P(A) + P(B|A′)·P(A′)

B can arrive through two doors. Send it down the A-lane and the A′-lane, then add. The shape exam Bayes problems are handed in.

Bayes — The FlipKap 4

P(A|B) = P(B|A)·P(A) / P(B)

Turn a known forward chance into its reverse. Numerator is the true-positive cell; denominator is every positive, from Oppsplitting.