In the 51% attack on the Proof Of Work Consensus Mechanism, it's often said that you need more hashing power than the rest of the network combined. Though I can't understand why you need that much.
If we break the blockchain network down in Alice, Bob, Charlie and Eve, and they all currently are honest. They all have 1/4 of the total hashing power. From what I understand, they all compete against eachother to calculate the right hash of a block, and the one winning the race gets its block on the blockchain. This happens one time each 10 minutes. Since they're all competing, the hash-power used to create a block in 10 minutes, is 1/4 of the total hashing power.
Now eve decides to go rogue, and builds her own blockchain. The difference is now, that she don't compete with anyone. That means that she can generate a block every 10 minutes (ish) on her private blockchain, keeping up to speed with the rest of the network. Having just a bit more power, she can then build blocks faster than Alice, Bob and Charlie combined, since they are "competing". At some point Eve's blockchain grows larger than the others, and she broadcasts it, and Alice, Bob and Charlie will per the rules switch to Eve's blockchain.
In this scenario the network was compromised by only using 25%(+1) of the hashing power. So why would Eve need 51%? Doesn't she just need to be faster than the fastest honest miner?
Hope you can tell me why I'm wrong.