  Hardy's paradox is of fundamental importance in quantum information theory.
So far, there have been two types of its extensions into higher dimensions: in
the first type the maximum probability of nonlocal events is roughly $9\%$ and
remains the same as the dimension changes (dimension-independent), while in the
second type the probability becomes larger as the dimension increases, reaching
approximately $40\%$ in the infinite limit. Here, we (i) give an alternative
proof of the first type, (ii) study the situation in which the maximum
probability of nonlocal events can also be dimension-independent in the second
type, and (iii) conjecture how the situation could be changed in order that
(ii) still holds.
