Math · Statistics and Probability

Probability formulas for JEE

Every Probability formula you need for JEE, grouped by concept.

FormulasRevision notes
17 formulas1 concepts
01Probability Basics & Events
01

Probability Basics & Events

17 formulas

Addition Theorem

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

Probability of occurrence of at least one of two events A or B.

additionunion

Addition Theorem (3 Sets)

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

Probability of occurrence of at least one of three events.

additionunion-three

Axiomatic Probability Sum

∑i=1nP(ωi)=1\sum_{i=1}^{n} P(\omega_i) = 1

The sum of probabilities of all elementary outcomes in a sample space equals 1.

applies whenAll ωi\omega_i form a mutually exclusive and exhaustive sample space.
axiomaticsample-space

Bayes' Theorem

P(Ei∣A)=P(Ei)P(A∣Ei)∑j=1nP(Ej)P(A∣Ej)P(E_i|A) = \frac{P(E_i) P(A|E_i)}{\sum_{j=1}^{n} P(E_j) P(A|E_j)}

Calculates the reverse conditional probability of partition E_i given outcome A.

applies whenPartitions EiE_i must be mutually exclusive and exhaustive, P(A)>0P(A) > 0.
bayesjee-advanced

Binomial Probability Distribution

P(X=r)=(nr)prqn−rP(X=r) = \binom{n}{r} p^r q^{n-r}

The probability of achieving exactly r successes in n independent Bernoulli trials.

applies whenIndependent trials, two possible outcomes (success/failure), constant success probability p, and p+q=1p+q=1.
binomial-distributionjee-advanced

Classical Definition of Probability

P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}

Probability of an event E evaluated using the count of favorable outcomes.

applies whenAll elementary outcomes in the sample space S must be equally likely.
classicalbasic

Complementary Event

P(A′)=1−P(A)P(A') = 1 - P(A)

Probability of the non-occurrence of event A.

complementbasic

Conditional Probability

P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}

Probability of event A occurring given that event B has already occurred.

applies whenP(B)>0P(B) > 0
conditionaljee-advanced

De Morgan's Law in Probability

P(A′∩B′)=1−P(A∪B)P(A' \cap B') = 1 - P(A \cup B)

Probability of neither event A nor event B occurring.

demorgancomplement

Difference of Events

P(A−B)=P(A∩B′)=P(A)−P(A∩B)P(A - B) = P(A \cap B') = P(A) - P(A \cap B)

Probability of occurrence of event A but not event B.

differenceset-theory

Probability of Exactly One Event

P(A∩B′)+P(A′∩B)=P(A)+P(B)−2P(A∩B)P(A \cap B') + P(A' \cap B) = P(A) + P(B) - 2P(A \cap B)

Probability that exactly one of the two events A or B occurs.

symmetric-differenceexact

Expected Value (Mean)

E(X)=∑ixipiE(X) = \sum_{i} x_i p_i

The mean or expected value of a discrete random variable X.

applies whenDiscrete random variable with ∑pi=1\sum p_i = 1.
random-variableexpectationjee-advanced

Independent Events Condition

P(A∩B)=P(A)P(B)P(A \cap B) = P(A) P(B)

The defining condition for two events to be statistically independent.

applies whenEvents A and B are strictly independent.
independencejee-advanced

Multiplication Theorem

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

Probability of the simultaneous occurrence of two dependent events.

applies whenP(A)>0P(A) > 0
multiplicationjee-advanced

Addition for Mutually Exclusive Events

P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)

Probability of occurrence of at least one of two mutually exclusive events.

applies whenA∩B=ϕA \cap B = \phi (The events are mutually exclusive).
additionmutually-exclusive

Law of Total Probability

P(A)=∑i=1nP(Ei)P(A∣Ei)P(A) = \sum_{i=1}^{n} P(E_i) P(A|E_i)

The total probability of an event A across all exhaustive partitions of the sample space.

applies whenPartitions EiE_i must be mutually exclusive and exhaustive.
total-probabilityjee-advanced

Variance of a Random Variable

Var(X)=∑ixi2pi−(E(X))2Var(X) = \sum_{i} x_i^2 p_i - (E(X))^2

The variance of a discrete probability distribution.

random-variablevariancejee-advanced
Other chapters
MathSets34 formulasMathRelations and Functions18 formulasMathTrigonometric Functions56 formulasMathComplex Numbers and Quadratic Equations36 formulasMathPermutations and Combinations21 formulasMathBinomial Theorem24 formulasMathSequence and Series25 formulasMathStraight Lines27 formulasMathConic Sections35 formulasMathIntroduction to Three-dimensional Geometry9 formulasMathLimits and Derivatives32 formulasMathStatistics25 formulasMathRelations and Functions14 formulasMathInverse Trigonometric Functions33 formulasMathMatrices44 formulasMathDeterminants23 formulasMathContinuity and Differentiability30 formulasMathApplications of Derivatives26 formulasMathIntegrals47 formulasMathApplications of the Integrals14 formulasMathDifferential Equations23 formulasMathVectors29 formulasMathThree-dimensional Geometry32 formulasMathProbability15 formulas

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