Why "double spent" can only happen if attacker controls 51% (or more) of the network compute power? As per many descriptions block hash is calculated based on it's headers and nonce field. The latter gives a chance to get the value that is less then a target by incrementing it. If it's a lottery why is it necessary for an attacker to have more compute power then the rest of the network? Can it have a chance to get ahead of main blockchain generation if it is just lucky? I.e. nonce's it uses for own blocks generation will be always giving him desired block hash too soon? Or is it just that probability is so low that we can safely assume that this never happen?
If it's a lottery why is it necessary for an attacker to have more compute power then the rest of the network? Can it have a chance to get ahead of main blockchain generation if it is just lucky?


Yes, a miner could attempt to 'get lucky' and pull of this attack with less than 51% of the hashing power. Keep in mind, if the attack fails, then the miner will have lost money attempting it. 51% is the point at which the miner should succeed on average. 

Note: mining is not a lottery by definition, it is only comparable to one in certain ways.