Answer to Question #86661 in Statistics and Probability for Bob

Question

The probability of an event A is 0.2. The probability of an event B is 0.5. If A and B are independent, then which of the following are true?

(I) P(A ∩ B) = 0.1
(II) P(A ∪ B) = 0.6
(III) P(A | B) = 0.2

Answer 1

Multiplication rule for two independent events:

When A and B are two independent events, then the combined probability of A and B is given by

$$P(A \cap B)=P(A)*P(B)$$ $$P(A \cap B)=0.2*0.5=0.1, True$$

For any two events A and B, the probability that either A or B will occur is given by the inclusion-exclusion rule 

$$P(A \cup B)=P(A)+P(B)-P(A \cap B)$$ $$P(A \cup B)=0.2+0.5-0.1=0.6, True$$

The conditional probability of A given B is 

$$P(A|B)=P(A \cap B)/P(B)$$

Event A is independent of B if the conditional probability of A given B is the same as the unconditional probability of A. That is, they are independent if 

$$P(A|B)=P(A)$$ $$0.2=0.2, True$$

Therefore, all three statements are true: (|) and (II) and (III).