Probability formula reference

Probability formulas organized by the event you need

Match not, or, and, given, or exactly k to the correct rule, then check independence, denominator, and equal-likelihood conditions before substituting.

Use the complement rule for not, the addition rule for a union, and the multiplication rule for an intersection. Conditional probability divides an intersection by the conditioning event. The independent multiplication shortcut and binomial formula require explicit model conditions; they are not default rules for every pair of events or repeated experiment.

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Probability formulas PDF

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Counting equally likely outcomes

Count the sample space only when every listed outcome has the same probability.

$P(A)=\frac{|A|}{|S|}$: Equally likely outcomesS is finite and nonempty. Every elementary outcome in S is equally likely. A is a subset of S.
Use this counting formula only for a finite sample space in which every elementary outcome is equally likely.
$\binom{n}{r}=\frac{n!}{r!(n-r)!}$: Combination countn is a nonnegative integer. r is an integer from zero through n. Order does not matter.
Use combinations to count selections of r objects from n distinct objects when order does not matter.

Complements and combined events

Use complements, unions, and intersections without double-counting overlap.

$P(A^c)=1-P(A)$: Complement ruleA is an event in a probability space. A superscript c contains exactly the outcomes not in A.
Use the complement rule when the probability of an event is easier to find from the probability that it does not occur.
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$: General addition ruleA and B are events in the same probability space.
Use the general addition rule to find the probability that at least one of two events occurs.
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$: Conditional probabilityA and B are events in the same probability space. The probability of B is greater than zero.
Use conditional probability when the sample space is restricted to outcomes in event B.
$P(A\cap B)=P(A\mid B)P(B)$: Multiplication ruleA and B are events in the same probability space. The probability of B is greater than zero for the conditional term shown.
Use the multiplication rule to find a joint probability from a conditional probability and its conditioning event.
$P(A\cap B)=P(A)P(B)$: Independent-event multiplicationA and B are independent events in the same probability space.
Use this shorter multiplication rule only after the model states or establishes that A and B are independent.

Repeated independent trials

Use the binomial model only when the trial count and success probability are fixed.

$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$: Binomial probabilityn is a nonnegative integer and k is an integer from zero through n. Trials are independent. Each trial has two outcomes and the same success probability p, where p is between zero and one inclusive.
Use the binomial formula for exactly k successes in n independent trials with one constant success probability.

Notation

Symbols used on this sheet

Interpret each symbol with the conditions attached to its formula.

Symbol	Meaning
$S$	The sample space containing all possible outcomes in the model.
$A^c$	The complement of A: outcomes in S that are not in A.
$A\cup B$	The union: A occurs, B occurs, or both occur.
$A\cap B$	The intersection: A and B both occur.
$P(A\mid B)$	The probability of A after restricting attention to event B.
$\binom nk$	The number of ways to choose k positions from n without regard to order.

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Short applications

These examples show when to choose a formula; detailed instruction belongs in the linked guide or calculator.

Union with overlap

Use the general addition rule because both events can occur.

P(A)=0.5, P(B)=0.4, P(A\cap B)=0.2

P(A\cup B)=0.5+0.4-0.2=0.7

Condition check: The disjoint A-only, both, and B-only regions are 0.3, 0.2, and 0.2, totaling 0.7.

Conditional probability

Restrict the denominator to the conditioning event B.

P(A\cap B)=0.18,\quad P(B)=0.30

P(A\mid B)=0.18/0.30=0.60

Condition check: Multiplying 0.60 by P(B) = 0.30 recovers the intersection 0.18.

Exactly two successes

Four independent constant-probability trials satisfy the binomial conditions.

X\sim\operatorname{Bin}(4,0.3)

P(X=2)=\binom42(0.3)^2(0.7)^2=0.2646

Condition check: Enumerating the six two-success position patterns gives the same factor of six.

Use conditions, not memory alone

Common formula confusions

Adding intersections

Using P(A) + P(B) for every union.

Fix: Subtract P(A intersection B) once unless the events are known to be mutually exclusive.

Assuming independence

Replacing P(A given B) with P(A) without evidence.

Fix: Keep the conditional multiplication rule unless independence is stated or established.

Using the whole sample space after conditioning

Dividing by 1 or by the original total for P(A given B).

Fix: Use P(B) or the count inside B as the new denominator.

Using binomial notation for changing probabilities

Applying one p when sampling without replacement changes the next success chance.

Fix: Use conditional multiplication or another appropriate model unless the binomial conditions truly hold.

Common questions

Probability formulas organized by the event you need FAQ

What is the difference between P(A and B) and P(A given B)?

P(A and B) is joint probability in the full sample space. P(A given B) is the fraction of B's probability that also lies in A, so it divides the joint probability by P(B).

When can I use P(A and B) = P(A)P(B)?

Only when A and B are independent for the model being used. The general rule is P(A and B) = P(A given B)P(B).

How do I find the probability of at least one event?

Often use a complement: one minus the probability that none occur. For two events, the general addition rule also gives P(A or B).

Why can a probability formula produce an impossible result?

The supplied values or assumptions may be inconsistent. Check that results remain between zero and one and obey intersection and union bounds.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

Probability Topics, OpenStax, Introductory Statistics 2e
Two Basic Rules of Probability, OpenStax, Introductory Statistics 2e

Written by Mathos AIPublished September 11, 2026