Attackers aren’t given infinite attempts, and even if they are, God doesn’t give them infinite time. So what you really want is to minimize the probability that the attacker guesses your password before giving up.
Suppose the attacking bot can make 200,000 attempts. By the first scheme, the probability the attacker guesses the password is .95 (plus an infinitesimal). By the first scheme but with a three-character password on a high roll, the probability is 1.00 (with 50 different characters, there are only 125K three-character words, so success in the remaining 199,999 attempts is certain).
By this measure, both passwords are weak, but the second is weaker than the first.
My Blackberry locks attackers out after 10 tries. So I would choose n=10 rather than n=200000. By that measure, the first scheme is roughly p=.950000, and the second is roughly p=.950072.
Taking it a bit further, these conditional probabilities depend upon some prior for various types of attacks. The numbers you have are P(guess password | password scheme AND attacker guesses via login attempts). They are both still really bad, but the difference is slightly greater if the password can be attacked by somehow exfiltrating a password hash database and running a billion guesses against that.
Attackers aren’t given infinite attempts, and even if they are, God doesn’t give them infinite time. So what you really want is to minimize the probability that the attacker guesses your password before giving up.
Suppose the attacking bot can make 200,000 attempts. By the first scheme, the probability the attacker guesses the password is .95 (plus an infinitesimal). By the first scheme but with a three-character password on a high roll, the probability is 1.00 (with 50 different characters, there are only 125K three-character words, so success in the remaining 199,999 attempts is certain).
By this measure, both passwords are weak, but the second is weaker than the first.
My Blackberry locks attackers out after 10 tries. So I would choose n=10 rather than n=200000. By that measure, the first scheme is roughly p=.950000, and the second is roughly p=.950072.
Taking it a bit further, these conditional probabilities depend upon some prior for various types of attacks. The numbers you have are P(guess password | password scheme AND attacker guesses via login attempts). They are both still really bad, but the difference is slightly greater if the password can be attacked by somehow exfiltrating a password hash database and running a billion guesses against that.