CivilForge

Addition Rule and Independence

P (A + B) = P (A) + P (B) - P (A, B), P (A, B) = P (A) P (B ∣ A)

p. 65 prints P(A+B) = P(A) + P(B) − P(A,B). Mutually exclusive means the intersection term is 0, so the union is the sum. p. 66 prints the joint P(A,B) = P(A) P(B|A). Independence means P(B|A) = P(B), so the joint equals the product. Those are different conditions. No separate complement equation is printed; P(A^c) = 1 − P(A) follows because A and A^c are mutually exclusive and cover the certain event. The probabilities below are original.

P (A), P (B)

Marginal probabilities

P (A, B)

Joint probability

P (B ∣ A)

Conditional probability

Step-by-step solved example

P(A) = 0.30, P(B) = 0.40, and P(A,B) = 0.05. Find P(A or B).

Original Venn: P(A)=0.30, P(B)=0.40, overlap 0.05. Union is 0.65, not 0.70 and not the independent product 0.12.

1. Union
P(A+B) = 0.30 + 0.40 − 0.05 = 0.65.
$0.65$
2. Not exclusive
If the intersection were 0, the union would be 0.70. Do not drop the 0.05.
3. Not independent
An independent joint would be 0.30×0.40 = 0.12. The given 0.05 is neither exclusive nor independent. P(B|A) = 0.05/0.30 = 0.1667.

Answer: 0.65

8 practice questions

0/8 correct

1.P(A)=0.30, P(B)=0.40, P(A,B)=0.05. P(A or B)?

2.If those events were mutually exclusive, the union would be

3.The independent joint P(A)P(B) would be

4.P(only A) = P(A) − P(A,B) equals

5.Mutually exclusive and independent are

6.P(B|A) = P(A,B)/P(A) equals

7.P(neither) = 1 − P(A+B) equals

8.The joint identity P(A,B)=P(A)P(B|A) is printed on