Question

Please help!!!

If P(A) + 0.40, P(B) = 0.75, and P(A and B) = 0.30, then what is true about the events A and B?
A) Event A and event B are mutually exclusive and are independent.
B) Event A and event B are mutually exclusive and are not independent.
C) Event A and event B are not mutually exclusive and are independent.
D) Event A and event B are not mutually exclusive and are not independent.
E) Event A and event B are not mutually exclusive, and independence cannot be determined with the information given.

Answer 1

D) Event A and event B are not mutually exclusive and are not independent.

Step-by-step explanation:

Since P(A and B) is not equal to 0, events A and B are not mutually exclusive.

P(A and B) = P(A) + P(B) - P(A or B)

Therefore,

P(A or B) = P(A) + P(B) - P(A and B) = P(A) + 0.75 - 0.3 = P(A) + 0.45

Since P(A or B) is a probability, it must be between 0 and 1:

0 ≤ P(A) + 0.45 ≤ 1

Subtracting 0.45 from each side, we get:

-0.45 ≤ P(A) ≤ 0.55

Since P(A) is a probability, it must also be between 0 and 1:

0 ≤ P(A) ≤ 1

Therefore,

0 ≤ P(A) ≤ 0.55

This means that A and B are not independent events.

The correct answer is D) Event A and event B are not mutually exclusive and are not independent.

Hope this helps!