I corrected the errors. The given probabilities are \(\frac{3}{4}\) and \(\frac{1}{2}\).

We do not know whether the events overlap. Therefore, we must establish the maximum and minimum bound of their overlap. The maximum overlap would be 1/2, which is the probability of event B (Assume A completely subsumes B). This is the maximum bound of P(A and B). On the other hand, if they do not overlap, meaning they are completely separate, P(A) and P(B) must add up to a number that is less than or equal to 1.

However, P(A) + P(B) = 3/4 + 1/2 = 5/4, which is more than 1. But probability of Event A or B cannot be more than 1. Therefore, there must be some overlap between them, which is: 5/4-1 = 1/4. This means, P(A and B) is at least 1/4. Earlier we established the maximum bound is 1/2. Therefore, 1/4 < P(A and B) < 1/2. Only C is within this range.