Luck & random is part of the game but not play a major role. 1 interviews/offers out of 5 schools applied - this is luck or random. 3 or more interviews/offers out of 5 school applied - this is the ability.
When preparing such a stressful 11+ exams, I certainly would not hang the hope on luck or other random factors, but well-prepared.
Apparently, schools have their own focus. In my own opinion, Dulwich is more Maths-focused school whereas Alleyn's & KCS are more on English. That means it's more likely you get interviews/offers from Dulwich if Maths is the children's strong subject but English is slightly weaker. That also explains why some children admit to school with scholarship but not offered by other schools.
@wildviking, ISEB exam format is different as it's computerized multi-choice. Not sure 13+, but with my 11+ experience, If your DC got Kings but not Alleyn's & Dulwich, I can only assume his English is much stronger than his Maths, as King's Maths is probably the easiest amongest the three schools (by looking at sample papers).
You can talk to the head teacher of his school to ask where he stands in the school and in the borough. In general, e.g. top 10%, top 15% or top 20%.
Also, you can take the mock tests to see where your child stands compare to children in other boroughs.
None of these gives you an exact estimation but it will be good enough.
Hope that helps.