For a 128-bit key, it would theoretically take many years and a percentage of the entire worlds power output to check every possible key, or for 256-bit it would take 3 sexdecillion years. But the brute force attack only continues to check while it has not found the key.
What are the chances that the key is the very last key for the brute force attack to check?
What are the chances that the key is one of the first few keys to be tested by the brute force attack? or even if it's one of the keys in the first 10% of all keys to be checked by the brute force attack? Just because there are a lot of zeros on all these numbers, couldn't the brute force attacker get lucky and guess the key?
Does the huge size of the search space (for 128-bit, 3.4*10^38) reduce that possibility to an acceptable risk level? People win the lottery all the time, despite there being a 1 in 10^6ish chance.
Has this ever been tested? Has anyone ever generated hundreds/thousands/n of keys and then tried to brute force them for a short time to see how often the key is found in minutes/days?
Edit: I am now curious about the distribution of keys, is it uniform? Also, can this reasoning be extended, say I do have thousands of similar security (all 128 aes) keys to crack, when I run a brute force attack to find them, a small amount of them should be found relatively quickly right? (or quickly compared to the theoretical max 10^100... years)