Probability

At a glance 11 formulas

1. 1Count the total first$P(E) = \dfrac{\text{favourable}}{\text{total}}$$\text{two dice: 36 outcomes; three coins: 8}$
2. 2Sums on two dice$\text{ways to roll a sum of } s = 6 - |s - 7|$$s = 2 \text{ to } 12;\ \text{a sum of 7 has the most ways, 6}$
3. 3Flip to the complement$P(\text{at least one}) = 1 - P(\text{none})$
4. 4Either or both$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$\text{neither} = 1 - P(A \cup B);\ \text{exclusive: add}$
5. 5Independent events$P(A \cap B) = P(A) \times P(B)$$\text{all must happen: multiply}$
6. 6Without replacementDraw one after another and shrink the fractions: two aces from a pack is $\tfrac{4}{52} \times \tfrac{3}{51}$, not $\tfrac{4}{52} \times \tfrac{4}{52}$.
7. 7Count selections both waysWhen order does not matter, count the favourable cases and the total both as selections ${}^{n}C_{r}$; mixing ordered and unordered counts is the classic slip.
8. 8Exactly r successes${}^{n}C_{r}\, p^{r} (1 - p)^{n - r}$$n \text{ tries};\ p = \text{chance of success on one try}$
9. 9Taking turns$P(\text{first wins}) = \dfrac{p}{1 - (1 - p)(1 - q)}$$p = \text{first player's chance per turn},\ q = \text{second's}$
10. 10Conditional probability$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$$\text{shrink the total to the outcomes where } B \text{ happened}$
11. 11Which source? (Bayes)When the result is known and the source is asked, find each route's chance (source, then result) and divide the route you want by the sum of all routes.