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Probability foundations - Exam P formula sheet

Every Exam P formula in probability foundations on one printable sheet, condensed from the 4 reference pages that teach them.

4 topics, 18 formula lines. Every line is condensed from the Exam P formula pages, which stay free to read in full. Print this page to get the sheet on paper; the site chrome drops away.

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Probability foundations

Axioms and sets

Axioms: $P (A) \geq 0, P (S) = 1, P (⋃_{i} A_{i}) = \sum_{i} P (A_{i}) for disjoint A_{i}$
Complement: $P (A^{c}) = 1 - P (A)$
Inclusion-exclusion (two sets): $P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Inclusion-exclusion (three sets): $P (A \cup B \cup C) = \sum P (A) - \sum P (A \cap B) + P (A \cap B \cap C)$
De Morgan: $(A \cup B)^{c} = A^{c} \cap B^{c}$

Counting

Multiplication rule: $n_{1} \times n_{2} \times \dots \times n_{k}$
Permutations: $P (n, k) = \frac{n !}{( n - k )!}$ order matters, no repetition
Combinations: $(k n) = \frac{n !}{k ! ( n - k )!}$ order does not matter
Multinomial: $\frac{n !}{k _{1} ! k _{2} ! \dots k _{m} !}$ split n items into labelled groups
Stars and bars: $(k - 1 n + k - 1)$ n identical items into k distinct boxes

Conditional probability

Definition: $P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) > 0$
Multiplication rule: $P (A \cap B) = P (A ∣ B) P (B)$
Independence: $P (A \cap B) = P (A) P (B) ⟺ P (A ∣ B) = P (A)$
Chain rule: $P (A_{1} \cap \dots \cap A_{n}) = P (A_{1}) P (A_{2} ∣ A_{1}) \dots P (A_{n} ∣ A_{1} \cap \dots \cap A_{n - 1})$

Bayes' theorem

Law of total probability: $P (A) = \sum_{k} P (A ∣ B_{k}) P (B_{k})$
Bayes: $P (B_{j} ∣ A) = \frac{P ( A ∣ B _{j} ) P ( B _{j} )}{\sum _{k} P ( A ∣ B _{k} ) P ( B _{k} )}$
Two-event form: $P (B ∣ A) = \frac{P ( A ∣ B ) P ( B )}{P ( A ∣ B ) P ( B ) + P ( A ∣ B ^{c} ) P ( B ^{c} )}$
Odds form: $\frac{P ( B _{1} ∣ A )}{P ( B _{2} ∣ A )} = \frac{P ( A ∣ B _{1} )}{P ( A ∣ B _{2} )} \cdot \frac{P ( B _{1} )}{P ( B _{2} )}$

Probability foundations - Exam P formula sheet. Formulas condensed from the free reference pages on this site. Not affiliated with or endorsed by the Society of Actuaries.