Question

Suppose that, given two events A and B you know the following three facts: - P(A)=0.4 - P(B)=0.5 - P(A and B)=0.2

Select each of the following statements below which must be true.
A. A and B are independent.
B. A and B are mutually exclusive
C. The probability of A or B occurring is equal to 0.7.
D. The probability of neither A nor B occurring is equal to 0.8.

Answer 1

Upon analyzing the conditions and using the principles of probability, statements A (A and B are independent) and C (The probability of A or B occurring is 0.7) are true, while statements B and D are false.

Step-by-step explanation:

Given two events A and B, we know the following probabilities:

• P(A) = 0.4
• P(B) = 0.5
• P(A AND B) = 0.2

Let's evaluate each statement:

A. A and B are independent.

Two events are independent if P(A AND B) = P(A)P(B). Here, P(A)P(B) would be 0.4 * 0.5 = 0.2, which equals P(A AND B). Therefore, statement A is true.

B. A and B are mutually exclusive.

Mutually exclusive events cannot occur at the same time, meaning P(A AND B) should be 0. Since P(A AND B) is 0.2, statement B is false.

C. The probability of A or B occurring is equal to 0.7.

The formula for the probability of A or B occurring is P(A) + P(B) - P(A AND B). So, 0.4 + 0.5 - 0.2 = 0.7. Hence, statement C is true.

D. The probability of neither A nor B occurring is equal to 0.8.

The probability of neither event occurring is 1 - P(A OR B), which we've established is 0.7. Therefore, 1 - 0.7 = 0.3, not 0.8. Consequently, statement D is false.