Compound And Conditional Probability Formula

1. If A and B are any two events then P( A ∩ B ) = P(B). P(A/B)

where B ≠ 0 and P(A/B) denotes the probability of occurrence of event A where B has already occurred.

2. If A and B are independent event then P(A ∩ B) = P(A).P(B)

(i). P(B/A) = P(A ∩ B ) / P(A) or P( A ∩ B) = P(A) . P(B/A).

(ii). Two events A and B are independent if and only if P( A ∩ B) = P(AB) = P(A).P(B)

(iii). If A and B are independent events then P(B/A) = P(B).

(iv). If A , B and C are any three independent events then –

P( A ∩ B ∩ C) = P[ A ∩ (B ∩ C) ] = P(A). P(B ∩ C) = P(A). (B).P(C).

(v). If A₁ , A₂ ,…………….A_n be any n events none of which is an impossible event then –

P( A₁ ∩ A₂ ∩…………….∩A_n ) = P(A₁) . P( A₂ /A₁).P(A ₃/A₁A₂)……..P(A_n/A₁, A₂ ,…………….A_n)

(vi). If A₁ , A₂ ,…………….A_n are independent events then

P( A₁ ∩ A₂ ∩…………….∩A_n )= P(A₁).P( A₂).P(A₃)……..P(A_n)

3. Complementation Rule : If A and B are two independent events then –

P( A U B) = 1 – P(A¹).P(B¹)

4. If A₁, A₂ ……….A_n are independent events then –

P( A₁ U A₂ U…………….U A_n ) = 1 – P(A¹₁) . P( A¹₂)……………….P(A¹_n)

5. The events A and φ are independent then –

P(A ∩ φ) = P(A) . P(φ).

6. The events A and S are independent then –

P( A ∩ S ) = P(A) = P(A). P(S).

7. If A and B be two non impossible mutually exclusive events then –

P( A U B ) = P(A) + P(B) .

8. If A and B be two non impossible independent events then –

P( A ∩ B) = P(A). P(B).

9. If A and B^I are independent events then –

P( A ∩ B^I ) = P(A). P(B^I).

10 . If A^I and B are independent events then –

P( A^I ∩ B) = P(A¹). P(B).

11 . If A^I and B^I are independent events then –

P( A^I ∩ B^I) = P(A¹). P(B^I).

12. If A and B are two events such that B ≠ 0 then –

P(A/B) + P(A^I/B) = 1

13. If A and B are two events such that A ≠ φ then –

P(B) = P(A). P(B/A) + P(A^I) . P(B/A^I)

Compound and conditional probability Formula