Probability Formulas

Complete Compilation (Class 11 & 12)

Event Type / Rule	Formula / Condition
Complementary Event (Not A)	$$ P(A’) = 1 – P(A) $$
Addition Theorem (A or B)	$$ P(A \cup B) = P(A) + P(B) – P(A \cap B) $$
Mutually Exclusive (Disjoint Events)	$$ A \cap B = \phi \implies P(A \cap B) = 0 $$ $$ P(A \cup B) = P(A) + P(B) $$
Exhaustive Events	If $$ E_1 \cup E_2 \cup … E_n = S $$: $$ \sum P(E_i) = 1 $$
Difference of Events (A but not B)	$$ P(A – B) = P(A) – P(A \cap B) $$ $$ = P(A \cap B’) $$

Conditional Probability
(Prob. of A given B has occurred)
$$ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) \neq 0 $$

Multiplication Theorem
(Prob. of A and B)
$$ P(A \cap B) = P(A) \cdot P(B|A) $$

Independent Events
(Occurrence of one does not affect the other)

Test Condition:
$$ P(A \cap B) = P(A) \cdot P(B) $$
Also implies: $$ P(A|B) = P(A) $$

Let $$ E_1, E_2, \dots, E_n $$ be a partition of sample space $$ S $$, and $$ A $$ be any event associated with $$ S $$.

Theorem of Total Probability
(Finding $$ P(A) $$ from parts)

$$ P(A) = \sum_{j=1}^{n} P(E_j) \cdot P(A|E_j) $$

Bayes’ Theorem
(Reverse Probability: Finding $$ P(E_i|A) $$)

$$ P(E_i|A) = \frac{P(E_i) \cdot P(A|E_i)}{\sum_{j=1}^{n} P(E_j) \cdot P(A|E_j)} $$

Concept	Formula
Mean / Expectation $$ E(X) $$ or $$ \mu $$	$$ \mu = \sum_{i=1}^{n} x_i p_i $$
Variance $$ \text{Var}(X) $$ or $$ \sigma^2 $$	$$ \sigma^2 = E(X^2) – [E(X)]^2 $$ $$ \sigma^2 = \sum x_i^2 p_i – (\sum x_i p_i)^2 $$
Standard Deviation	$$ \sigma = \sqrt{\text{Var}(X)} $$
Binomial Distribution ($$ n $$ trials, $$ r $$ successes)	$$ P(X=r) = {}^nC_r p^r q^{n-r} $$ Where $$ p + q = 1 $$