The probability that both events X and Y occur is 3/8, they are not mutually exclusive. The probability that neither event occurs is 3/14, and the probability that only one event occurs is 25/56.
The question involves calculating the probability of events occurring individually and in combination. Given that P(X)=4/7, P(Y)=1/2, and the conditional probability P(X | Y)=3/4, we can find the probability that both events occur using the formula P(X and Y) = P(X | Y) × P(Y). Substituting the given values, we get P(X and Y) = (3/4) × (1/2) = 3/8. Thus, the probability that both events X and Y occur is 3/8.
To determine if events X and Y are mutually exclusive, we consider the definition of mutually exclusive events, which means that P(X and Y) = 0. Since our computed probability P(X and Y) is 3/8, which is not zero, events X and Y are not mutually exclusive.
Next, we determine the probability that both events do not occur. This is found by calculating P(not X and not Y) = (1 - P(X)) × (1 - P(Y)), which gives P(not X and not Y) = (1 - 4/7) × (1 - 1/2) = 3/7 × 1/2 = 3/14.
Finally, to find the probability that only one of the events occurs, we use the formula P(X or Y) - P(X and Y), where P(X or Y) can be found using P(X) + P(Y) - P(X and Y). Since P(X or Y) = 4/7 + 1/2 - 3/8, the probability that only one of the events occurs is 4/7 + 1/2 - 3/8 = 25/56.