Expected value in pool mining vs. solo mining

Question

Set aside the fact that pool mining provides a more steady stream of income than solo mining, and the fact that pools charge a fee for their services. Disregarding those facts, I have an argument for why pool mining should carry higher expected value. This topic has been covered before and apparently I'm wrong:

"...solo mining and pooled mining should, over the long term, produce precisely the same revenue"

Here's my argument, please break it and show me why it's wrong:

Alice, Bob, Cecil and Danny compete to find the next block. They know it's in one of 10 boxes. Alice and Bob are solo miners and randomly search through boxes one by one. Cecil and Danny also search through boxes randomly, but they are in the same pool, so they decide that Cecil will search through boxes 1-5 and Danny will search through boxes 6-10. This way they will never waste computing power searching the same box twice. Alice and Bob on the other hand don't coordinate with each other, so sometimes Alice will search in a box that Bob already knows is empty. How are Alice (solo) and Danny (pool) equally likely to find the block?

Answer 1

You are assuming that Cecil and Danny can collaborate to avoid duplicating work. They can't, at least not in any meaningful way. In your example, there are a small number of boxes for them to search. In Bitcoin, there are trillions of boxes. Any collaboration about keeping track of who searched what would take much more work than everybody searching randomly and accepting the extremely small probability of checking the same thing twice.