Probability of Success in Slotted Medium Access
There is a clear explanation given in these lecture notes MIT course Digital communication systems (read by H. Balakrishnan and G. Verghese).
The specific part that you require is this&gt;

If each node sends with probability $p$, then the probability that exactly one node sends in any given slot is $Np(1 − p)^{N−1}$.
The reason is that the probability that a specific node sends in the time slot is $p$, and for its transmission to be successful, all the other nodes should not send. That combined probability is $p(1 − p)^{N−1}$. Now, we can pick the successfully transmitting node in $N$ ways, so the probability of exactly one node sending in a slot is $Np(1 − p)^{N−1}$.