An event X can happen with probability p and event Y can happen...

Given $P(X) = p$ and $P(Y) = q$.

Since $X$ and $Y$ are independent events, $P(X \cap Y) = P(X)P(Y) = pq$.

The probability that exactly one of the events happens is given by $P(X \cap Y') + P(X' \cap Y) = P(X) + P(Y) - 2P(X \cap Y) = p + q - 2pq$.

Statement I claims this probability is $p + q - pq$, which is incorrect.

The probability that at least one of the events happens is given by $P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) = p + q - pq$.

Statement II claims this probability is $p + q - 2pq$, which is also incorrect.

Therefore, neither Statement I nor Statement II is correct.

Answer: Neither I nor II