Question

Let a and b be any two events. Which of the following statements, in general, are false?

1) Statement 1: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
2) Statement 2: P(A ∩ B) = P(A) * P(B)
3) Statement 3: P(A') = 1 - P(A)
4) Statement 4: P(A ∩ B') = P(A) - P(A ∩ B)

Answer 1

Statement 2 is generally false unless A and B are independent, and Statement 4 is false. They incorrectly assume the nature of the relationship between events A and B.

Step-by-step explanation:

When assessing the truth of the given statements about probability, we need to understand that they refer to the basic rules of probability concerning the relationship between two events, A and B, in the context of a probability space.

• Statement 1: This statement is true and represents the addition rule for any two events A and B. The probability of the union of A and B is equal to the sum of the probabilities of A and B, minus the probability of their intersection.
• Statement 2: This statement is generally false unless A and B are independent events. The probability of the intersection of A and B is equal to the product of their probabilities only if A and B are independent.
• Statement 3: This statement is true and represents the complement rule. The probability of the complement of event A (not A) is equal to 1 minus the probability of A.
• Statement 4: This statement is false. The probability of A intersecting the complement of B is not the same as the probability of A minus the probability of A and B. This would only be true if A and B are mutually exclusive, which is not assumed here.

Overall, the false statements are Statement 2 and Statement 4, as they make assumptions about the nature of the events that are not necessarily true.