Question

Given that P(A) + P(B) 0.9, P(AB) = 0.5, and P(BA) 0.4.

(a) find P(A) and P(B);
(b) are A and B disjoint?
(c) are A and B independent?

Answer 1

To find P(A) and P(B), we can use the formula: P(A) + P(B) = P(A AND B) + P(A OR B). Given that P(A) + P(B) = 0.9 and P(AB) = 0.5, we can substitute the values and solve for P(A) and P(B). A and B are not disjoint because P(AB) and P(BA) are not equal, and A and B are not independent because P(B|A) and P(B) are not equal.

Step-by-step explanation:

To find P(A) and P(B), we can use the formula:

P(A) + P(B) = P(A AND B) + P(A OR B)

Given that P(A) + P(B) = 0.9 and P(AB) = 0.5, we can substitute the values:

0.9 = 0.5 + P(A OR B)

P(A OR B) = 0.4

Since A and B are mutually exclusive, the probability of A OR B is the sum of their individual probabilities: P(A OR B) = P(A) + P(B

Therefore, we have the equation:

0.4 = P(A) + P(B)

Substituting this equation into the first equation, we get:

0.9 = 0.5 + (0.4)

P(A) + P(B) = 0.5

Since A and B are mutually exclusive, P(A) and P(B) are equal.

Therefore, P(A) = P(B) = 0.25

So, the answer for part (a) is P(A) = P(B) = 0.25.

For part (b), two events A and B are disjoint if their intersection (A AND B) is empty. In this case, since P(AB) = 0.5 and P(BA) = 0.4, the intersection is not empty.

Therefore, A and B are not disjoint.

For part (c), two events A and B are independent if P(A|B) = P(A) and P(B|A) = P(B). In this case, P(B|A) = 0.4 and P(B) = 0.25, which are not equal.

Therefore, A and B are not independent.