Question

1. let a and b be two events. suppose that p(a)=0.2, p(b)=0.2, p(a u b)=0.4. which of the following statements is true? a) the probabilities of a and b are impossible. b) a and b are mutually exclusive (disjoint). c) a and b are independent. d) a and b are both mutually exclusive and independent. e) none of the other statements are correct.

Explanation:

Step1: Recall probability formula

We know that \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Given \(P(A) = 0.2\), \(P(B)=0.2\) and \(P(A\cup B)=0.4\). Substitute these values into the formula: \(0.4=0.2 + 0.2-P(A\cap B)\).

Step2: Solve for \(P(A\cap B)\)

Simplify the right - hand side of the equation: \(0.2 + 0.2-P(A\cap B)=0.4 - P(A\cap B)\). Since \(0.4=0.4 - P(A\cap B)\), we get \(P(A\cap B)=0\).

Step3: Determine relationship between A and B

If \(P(A\cap B) = 0\), then events A and B are mutually exclusive (disjoint) because mutually exclusive events satisfy the condition that the probability of their intersection is zero.

Answer:

B. A and B are mutually exclusive (disjoint).