Question

For which of the following probability assignments are events A and B independent?

A. P(A∩Bc)=0.3, P(A∩B)=0.12, and P(Ac∩B)=0.4
B. P(A∩Bc)=0.3, P(A∩B)=0.3, and P(Ac∩B)=0.3
C. P(A∩Bc)=0.1, P(A∩B)=0.1, and P(Ac∩B)=0.4
D. P(A∩Bc)=0.3, P(A∩B)=0.0, and P(Ac∩B)=0.2
E. P(A∩Bc)=0.5, P(A∩B)=0.1, and P(Ac∩B)=0.4

Answer 1

To determine if events A and B are independent, compare the probabilities of their intersections and individual probabilities. Only option B, with P(A∩Bc) = 0.3, P(A∩B) = 0.3, and P(Ac∩B) = 0.3, shows events A and B to be independent.

Step-by-step explanation:

To determine if events A and B are independent, we need to compare the probabilities of their intersections and individual probabilities. If the probabilities of the intersections match the product of the individual probabilities, then the events are independent.

In option A, P(A∩Bc) = 0.3, P(A∩B) = 0.12, and P(Ac∩B) = 0.4. However, P(A∩B) = 0.12 ≠ P(A∩Bc) * P(B).

In option B, P(A∩Bc) = 0.3, P(A∩B) = 0.3, and P(Ac∩B) = 0.3. Here, P(A∩B) = 0.3 = P(A∩Bc) * P(B), so events A and B are independent.

In option C, P(A∩Bc) = 0.1, P(A∩B) = 0.1, and P(Ac∩B) = 0.4. However, P(A∩B) = 0.1 ≠ P(A∩Bc) * P(B).

In option D, P(A∩Bc) = 0.3, P(A∩B) = 0.0, and P(Ac∩B) = 0.2. Since P(A∩B) = 0.0, events A and B are independent regardless of the other probabilities.

In option E, P(A∩Bc) = 0.5, P(A∩B) = 0.1, and P(Ac∩B) = 0.4. However, P(A∩B) = 0.1 ≠ P(A∩Bc) * P(B).