Q: if A and B are independent events, how many of the following statements must be true?


1. A' and B are independent
2. A and B' are independent
3. A' and B' are independent

is choice 3 the only one that is true?

I don't get the question. What are A' and B'?

To determine if the statements are true, we need to understand the concept of independence between events. Two events A and B are considered independent if the occurrence (or non-occurrence) of one event does not affect the probability of the other event.

Now, let's evaluate each statement one by one:
1. A' and B are independent: This statement suggests that the complement of event A and event B are independent. However, this is not necessarily true. The complement of A, denoted as A', is the event that A does not occur. The independence between A' and B does not guarantee independence between A and B. Therefore, statement 1 is not necessarily true.

2. A and B' are independent: Similarly, this statement suggests that event A and the complement of event B, denoted as B', are independent. Like in statement 1, the independence between A and B' does not ensure independence between A and B. So, statement 2 is also not necessarily true.

3. A' and B' are independent: This statement suggests that the complement of both events A and B, namely A' and B', are independent. Surprisingly, this statement is true. If A and B are independent events, it follows that A' and B' are also independent. This is because if the occurrence of A doesn't affect B, then by the same logic, the non-occurrence of A (i.e., A') should also not affect B. Thus, statement 3 is indeed true.

In summary, only statement 3 is necessarily true when A and B are independent events.