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Probability Theory Formulas

Probability Theory Formula Reference

Complete mathematical reference for probability theory fundamentals: random experiments, events, probability definitions, and classical probability models

Foundation FormulasPractical ApplicationsMathematical Rigor

Basic Probability Definitions

Fundamental probability definitions and calculation methods

Basic Probability Definitions
Essential mathematical formulas and relationships

Classical Probability

P(A)=∣A∣∣Ω∣=favorable outcomestotal outcomesP(A) = \frac{|A|}{|\Omega|} = \frac{\text{favorable outcomes}}{\text{total outcomes}}

For equally likely outcomes in finite sample space

Statistical Probability

P(A)=lim⁡n→∞nAnP(A) = \lim_{n \to \infty} \frac{n_A}{n}

Frequency approach as number of trials approaches infinity

Geometric Probability

P(A)=measure of Ameasure of ΩP(A) = \frac{\text{measure of A}}{\text{measure of } \Omega}

For continuous sample spaces with uniform distribution

Complement Rule

P(Ac)=1−P(A)P(A^c) = 1 - P(A)

Probability of complement event

Probability Axioms & Properties

Kolmogorov's axioms and fundamental probability properties

Probability Axioms & Properties
Essential mathematical formulas and relationships

Non-negativity Axiom

P(A)≥0 for all events AP(A) \geq 0 \text{ for all events } A

All probabilities are non-negative

Normalization Axiom

P(Ω)=1P(\Omega) = 1

Probability of sample space equals 1

Countable Additivity

P(⋃i=1∞Ai)=∑i=1∞P(Ai) if Ai disjointP(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i) \text{ if } A_i \text{ disjoint}

For mutually exclusive events

Addition Formula

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

General addition rule for any two events

Monotonicity

If A⊂B, then P(A)≤P(B)\text{If } A \subset B, \text{ then } P(A) \leq P(B)

Probability is monotonic with respect to set inclusion

Sub-additivity

P(⋃i=1∞Ai)≤∑i=1∞P(Ai)P(\bigcup_{i=1}^{\infty} A_i) \leq \sum_{i=1}^{\infty} P(A_i)

Union probability bounded by sum of individual probabilities

Event Operations & Relationships

Set operations on events and their probability implications

Event Operations & Relationships
Essential mathematical formulas and relationships

Union (OR)

A∪B={ω:ω∈A or ω∈B}A \cup B = \{\omega : \omega \in A \text{ or } \omega \in B\}

Event that at least one of A or B occurs

Intersection (AND)

A∩B={ω:ω∈A and ω∈B}A \cap B = \{\omega : \omega \in A \text{ and } \omega \in B\}

Event that both A and B occur

Difference

A−B=A∩Bc={ω:ω∈A and ω∉B}A - B = A \cap B^c = \{\omega : \omega \in A \text{ and } \omega \notin B\}

Event that A occurs but B does not

De Morgan's Laws

A∪B‾=A‾∩B‾,A∩B‾=A‾∪B‾\overline{A \cup B} = \overline{A} \cap \overline{B}, \quad \overline{A \cap B} = \overline{A} \cup \overline{B}

Complement of union equals intersection of complements

Disjoint Events

A∩B=∅⇒P(A∪B)=P(A)+P(B)A \cap B = \emptyset \Rightarrow P(A \cup B) = P(A) + P(B)

Mutually exclusive events have additive probabilities

Conditional Probability & Independence

Conditional probability formulas and independence relationships

Conditional Probability & Independence
Essential mathematical formulas and relationships

Conditional Probability

P(A∣B)=P(A∩B)P(B),P(B)>0P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0

Probability of A given that B has occurred

Multiplication Rule

P(A∩B)=P(A∣B)⋅P(B)=P(B∣A)⋅P(A)P(A \cap B) = P(A|B) \cdot P(B) = P(B|A) \cdot P(A)

Joint probability as product of conditional and marginal

Chain Rule

P(A1∩A2∩⋯∩An)=P(A1)P(A2∣A1)P(A3∣A1∩A2)⋯P(An∣A1∩⋯∩An−1)P(A_1 \cap A_2 \cap \cdots \cap A_n) = P(A_1)P(A_2|A_1)P(A_3|A_1 \cap A_2) \cdots P(A_n|A_1 \cap \cdots \cap A_{n-1})

General multiplication rule for multiple events

Total Probability Formula

P(B)=∑i=1nP(B∣Ai)P(Ai)P(B) = \sum_{i=1}^n P(B|A_i)P(A_i)

For partition {A₁, A₂, ..., Aₙ} of sample space

Bayes' Theorem

P(Ai∣B)=P(B∣Ai)P(Ai)∑j=1nP(B∣Aj)P(Aj)P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^n P(B|A_j)P(A_j)}

Posterior probability formula

Independence

P(A∩B)=P(A)⋅P(B)⇔P(A∣B)=P(A)⇔P(B∣A)=P(B)P(A \cap B) = P(A) \cdot P(B) \Leftrightarrow P(A|B) = P(A) \Leftrightarrow P(B|A) = P(B)

Events A and B are independent

Bernoulli Trials & Binomial Model

Formulas for Bernoulli processes and binomial probability models

Bernoulli Trials & Binomial Model
Essential mathematical formulas and relationships

Bernoulli Trial

P(X=k)={pif k=11−pif k=0P(X = k) = \begin{cases} p & \text{if } k = 1 \\ 1-p & \text{if } k = 0 \end{cases}

Single trial with success probability p

Binomial Probability

P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

k successes in n independent Bernoulli trials

Binomial Coefficient

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Number of ways to choose k items from n items

Binomial Mean

E[X]=npE[X] = np

Expected number of successes

Binomial Variance

Var(X)=np(1−p)\text{Var}(X) = np(1-p)

Variance of binomial distribution

Bernoulli Properties

E[X]=p,Var(X)=p(1−p)E[X] = p, \quad \text{Var}(X) = p(1-p)

Mean and variance of single Bernoulli trial

🎯 Applications of Probability Theory Formulas

Real-world applications where probability theory formulas are essential

Medical Diagnosis

Bayes' theorem for calculating disease probability given test results, considering sensitivity and specificity.

Quality Control

Binomial model for defect rates, classical probability for sampling without replacement.

Risk Assessment

Conditional probability for event dependencies, total probability formula for comprehensive risk analysis.

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