Determine whether the following statements are true or false:

1. P(A and B)=P(B|A)P(B).
2. When A and B are independent, P(A)=P(B).
3. When A and B are independent, P(A|B)=P(A).
4. When A includes B, P(A|B)=1.
5. When B is the complement of A, P(A or B)=1.
6. When B is the complement of A, P(A and B)=1.

What do you think?

To determine whether the statements are true or false, we'll use the general properties and definitions of probability.

1. P(A and B) = P(B|A)P(A) (This is the Multiplication Rule of Probability.)
- False. The correct formula is P(A and B) = P(A|B)P(B). The conditional probability P(B|A) is not equal to P(B), but rather relates to the probability of B given that A has occurred.

2. When A and B are independent, P(A) = P(B).
- False. When A and B are independent events, it means that the occurrence or non-occurrence of one event does not affect the probability of the other event. However, it does not mean that their probabilities are necessarily equal. In general, P(A) ≠ P(B) for independent events unless they have the same probability to begin with.

3. When A and B are independent, P(A|B) = P(A).
- True. When A and B are independent events, the probability of A occurring given that B has occurred (P(A|B)) is equal to the unconditional probability of A (P(A)).

4. When A includes B, P(A|B) = 1.
- False. If A includes B, then P(A|B) is not necessarily 1. The probability of A occurring given that B has occurred can be 1 if A is certain to happen whenever B happens, but it depends on the specific circumstances. In general, it cannot be assumed that P(A|B) = 1 solely based on A including B.

5. When B is the complement of A, P(A or B) = 1.
- True. If B is the complement of A, it means that either A or B must occur (but not both) because they are mutually exclusive. In this case, P(A or B) = 1, representing that at least one of A or B will occur.

6. When B is the complement of A, P(A and B) = 1.
- False. If B is the complement of A, it means that they cannot both occur simultaneously. Therefore, the probability of both A and B occurring (P(A and B)) is always 0, as they are mutually exclusive events.