What must be true for p (a and b) = p (a) * p (b)?

a. a and b cannot be disjoint.
b. a and b must be independent.
c. a and b cannot be independent.
d. a and b must be disjoint.
e. it is always true.

option B. A and B must be independent.

Explanation:

1) Disjoint events do not have any outcomes in common. If two events are disjoint means that if one happens the other cannot happen.

Since two disjoint events cannot happen simultaneously, A∩B = ∅ and P (A and B) = 0.

2) Independent events are those for which knowing whether or not one the events occurs does not change the occurrence (probability) of the other or others event.

Event B is independent of Event A if: P(B | A) = P(B)

The probability of ocurrence of two indepent events A and B is calculated by multiplying the individual events, that is the probability of A and B equals the probability of A times the probability of B. In symbols, P(A and B) = P(A) × P(B).

Then, the option B, reflects a property of or rule for independent events: when A and B are indepentent P (A and B) = P (A) × P (B).

3) Two or more events cannot be disjoint and independent. Since, in the case of disjoint events, the occurrence of one of the events means that the other events cannot happen, the occurrence of one event affects the probability of the other events, and they are not independent.

Nevertheless, since A and B cannot not be both true for disjoint events, P(A and B) is 0. And, since P(A) or P(B) is equal to zero, it results that P(A and B) = P(A)×P(B) = 0.

4) Conclusion: P (A and B) = P (A) × P (B) is the general case rule for independent events, but it is also true for disjoint events, for which P (A and B) is always 0.