I am trying to understand Satoshi's paper . On page 6 of the paper, he calculates the probability that someone can attack the blockchain from z blocks behind. He begins by defining:
My question: The mining of a block i.e., solving the hash puzzle is a completely brute-force trial-and-error process in which one keeps trying a random nonce until one gets a hit. No one has an upper hand in this process. So shouldn't p and q be equal to each other?
EDIT: since comment would have been very long. Responding to the comments, in that case it gets even worse. q can be greater than p. Let the Bitcoin network be composed of M mining pools with their compute power be given by p_i. Then all an attacker needs to do is to form a pool whose compute power is greater than max(p_i). Am I missing something? In other words, the pool with max compute power can attack the blockchain already.
In an ideal design, if I want to attack the blockchain, I should be pitted against the sum of p_i not the max of p_i.