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.


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