Probability Calculator

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Formulas and Derivations

Probability rule
P(E) = n(E) / n(S)
Complement rule
P(A) = 1 - P(A')
Addition rule
P(A∪B) = P(A) + P(B) - P(A∩B)
Conditional probability formula
P(A|B) = P(A∩B) / P(B)
Multiplication rule
P(A∩B) = P(B) * P(A|B)
(Rearrangement of the conditional probability formula)
Independent events formula
P(A∩B) = P(A) * P(B)
Formula for finding P(A) from
P(B) and P(A∪B) (for
independent events)
P(A) = (P(A∪B) - P(B)) / (1 - P(B))
Since P(A) = P(A|B') (as A and B are independent)
= P(A∩B') / P(B')
= (P(A∪B) - P(B)) / (1 - P(B))
Bayes' Theorem
P(A|B) = P(B|A) * P(A) / P(B)
Since P(A|B) = P(A∩B) / P(B)
= (P(A∩B) * P(A)) / (P(A) * P(B)))
= P(A∩B)/P(A) * P(A)/P(B)
= P(B|A) * P(A) / P(B)