# Throughput of slotted ALOHA when one node goes out of sync

What would be the throughput of slotted ALOHA when one node goes out of sync? Closer to ALOHA or slotted ALOHA?

According to my calculations:

For ALOHA we have probability of success = $$N \cdot p \cdot (1 - p)^{N - 1}$$
For slotted ALOHA we have probability of success = $$N \cdot p \cdot (1 - p)^{2(N - 1)}$$
For one node going out of sync probability of success = $$N \cdot p \cdot (1 - p)^{N}$$

According to my intuition it should be almost equal to slotted ALOHA as for large N, $$lim (N - 1) \rightarrow N$$ and these two probabilities should become equal.

I have had a look at this derivation but didn't completely understand the formal proof for the efficiency of slotted ALOHA.