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Probability Formulas
Browse 3 reviewed probability formulas with concrete inputs, outputs, calculations, applicability checks, and LaTeX.
Probability
Bayes’ Theorem
P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}Updates the probability of an event after observing evidence.
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Binomial Probability Formula
P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}Finds the probability of exactly k successes in n independent Bernoulli trials.
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Conditional Probability
P(A\mid B)=\frac{P(A\cap B)}{P(B)}Calculates the probability of A when B is known to have occurred.
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Probability
Binomial Probability Mass Function
P(X=k)=\binom nkp^k(1-p)^{n-k}Computes exact successes in independent Bernoulli trials.
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Combination Formula
\binom{n}{r}=\frac{n!}{r!(n-r)!}Counts unordered selections of r items from n distinct items.
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Discrete Expected Value
\mathbb E[X]=\sum_x xP(X=x)Computes the probability-weighted average of a discrete random variable.
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Geometric Distribution PMF
P(X=k)=(1-p)^{k-1}pModels the trial number of the first success.
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Law of Total Probability
P(A)=\sum_i P(A\mid B_i)P(B_i)Combines conditional probabilities across a partition.
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Law of Total Probability
P(B)=\sum_iP(B\mid A_i)P(A_i)Computes an event probability over a partition of cases.
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Normal Distribution Density
f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}Defines the bell-shaped normal probability density.
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Permutation Formula
P(n,r)=\frac{n!}{(n-r)!}Counts ordered selections of r items from n distinct items.
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Poisson Probability Mass Function
P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}Models counts of independent events in a fixed interval.
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Probability Addition Rule
P(A\cup B)=P(A)+P(B)-P(A\cap B)Finds the probability of at least one of two events.
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Probability Complement Rule
P(A^c)=1-P(A)Finds the probability that an event does not occur.
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Variance Identity
\operatorname{Var}(X)=\mathbb E[X^2]-\mathbb E[X]^2Expresses variance using first and second moments.
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