4.2.2 problem 24

One way to think about the problem is that the event $X<r$ counts all sequences of $n$ independent Bernoulli trials, where the number of failures is larger than $n-r$. If we extend the number of trials indefinitely, this implies that more than $n-r$ failures occured before the $r$-th success, because otherwise, we’d have $X\ge r$. The probability of this event is $P(Y>n-r)$.

Implication in the reverse direction can be shown analogously.