Probability rules
- Read it, not yetReach the end of the page
- Practice it, not yetAnswer a few questions
- Master it, not yet80% on a drill of 8 or more
Or: add, then subtract the overlap. And: multiply. Given: count only inside what you were told. Mutually exclusive: can't both happen. Independent: one says nothing about the other.
Why it matters
Probability is honest bookkeeping: list what can happen, then ask what share of it is the thing you care about.
Many costly mistakes come from mixing up P(A given B) and P(B given A).
- MedicineA 99% accurate test for something rare is often wrong when it says yes.
- FraudIf fraud is rare, most flagged transactions are honest.
- EverydayFive separate 20% risks: the chance at least one happens is 67%, not 100%.
Tennis example
Break-point conversion is a conditional probability: P(win the point | break point). Count only the break points. Convert 20 of 50 and that is 40%.
Coin: P(heads) = 1/2
Flips 0
Flip it. The dark line marks the true 50%.
Two heads in a row (and means multiply): 0 of 0 pairs. True odds 0.5 × 0.5 = 25%.
Die: each face 1/6 (16.7%)
Rolls 0
Turn this on to watch P(2 | even) head toward 1/3.
Check yourself
Two coins and a die: how many outcomes?
242 × 2 × 6.One die roll: P(not a 1 or 2)?
4/61 − 2/6.A = {1, 2, 3}, B = {3, 4} on one die roll. P(A or B)?
4/63/6 + 2/6 − 1/6.
The trap
The mistake people make most on this one.
Mutually exclusive events are not independent. If A happens B cannot, so knowing A changes B completely.
Formulas
5 formulas and a way to remember them.
- Basics
- 0 ≤ P(A) ≤ 1, all outcomes add to 1, P(not A) = 1 − P(A)
- Or
- P(A or B) = P(A) + P(B) − P(A and B)
- Given
- P(A | B) = P(A and B) / P(B)
- And
- P(A and B) = P(A) × P(B | A); if independent, P(A) × P(B)
- Tree
- Multiply along a branch, add branches with the same ending
- Remember it
- Or: two guest lists, count overlaps once. And: a combo lock, every dial must hit. Given: a filter, then count.
Watch and go deeper
3 free videos and pages.
- StatQuest: Conditional Probabilities, Clearly ExplainedConditional probability built from a simple table of counts. About 11 minutes.
- 3Blue1Brown: Bayes theorem, the geometry of changing beliefsBayes' rule drawn as areas, with the classic base rate example. Pause at the example and answer it first. About 15 minutes.
- Seeing Theory: Compound ProbabilitySet theory, counting and conditional probability, all interactive. About 15 minutes.