Question

Assume the following probabilities for two events, A and B: P(A) = 0.50,P(B) = 0.50, and P(A ∪ B) = 0.85. Then we can conclude that

• A. A and B are mutually independent
• B. A and B are equally likely
• C. A and B are not mutually independent
• D. A and B are mutually exclusive

Answer 1

The Correct Option is A

To determine the relationship between events A and B, we need to check if they are mutually independent. Two events A and B are considered independent if:

P(A ∩ B) = P(A) * P(B)

We are provided with the following probabilities:

• P(A) = 0.50
• P(B) = 0.50
• P(A ∪ B) = 0.85 

Using the principle of inclusion-exclusion for probabilities, we know:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Substituting the given values:

0.85 = 0.50 + 0.50 - P(A ∩ B)

0.85 = 1.00 - P(A ∩ B)

P(A ∩ B) = 1.00 - 0.85

P(A ∩ B) = 0.15

Next, we calculate P(A) * P(B):

P(A) * P(B) = 0.50 * 0.50 = 0.25

Since P(A ∩ B) = 0.15, which is not equal to P(A) * P(B) = 0.25, events A and B are not mutually independent.

The correct conclusion is: A and B are not mutually independent.