Question

Suppose a computer's login mechanism first asks for a user name, and then for the password, regardless of whether or not the login name is that of a valid user. Herman the Hacker has a list of 10,000 commonly used login names, and a list of 100,000 commonly used passwords. On this computer, there are 100 accounts that have both a login name on the first list and a password on the second list. For simplicity, assume that no other accounts have either a login name or password on these lists.

Herman has written code that can attempt 1000 logins per second on this computer. This program repeatedly chooses a random login name from the first list and a random password from the second list. Compute the expected number of seconds it will take Herman to break into some account on the machine, and identify that number from the list below.

Hint: Since names and passwords are chosen randomly, repetitions are possible. It may indeed be easier to forget what names/passwords have been tried than to keep a list of what has been tried and avoid duplicate attempts. As a result, there is an easy formula for the expected number of tries. If the probability of success on any given try is p, then the expected number of tries until the first success is exactly 1/p.

A. 1.1
B. 100.1
C. 10,000
D. 1000.1

(Thank you for the help!)

Answer 1

Here we have 10,000 login names among them we can assume 100 are correct as the question suggest .

So,

the probablity of getting the correct login name is P_{​​​​​​1} = 100 ÷ 10,000 = 0.01

Similarly Among 100,000 password we can assume only 100 are correct

So,

the probability of getting correct password is. P ₂ = 100 ÷ 100,000 = 0.001

As we have

If the probability of success on any given try is p, then the expected number of tries until the first success is exactly 1 ÷ P.

So,

expected number of tries when we get the first success in login name is = 1÷ P1_{​​​​​1} = 1 ÷ 0.01 = 100

Similarly for password : 1÷ P₂ = 1 ÷ 0.001 =1000

So we have to do 100+1000 =1100 number of tries to get the first login name and password

As,

Henrietta has written code that can attempt 1000 logins per second on this computer.

So

the expected number of seconds it will take Henrietta to break into some account on the machine is 1100 ÷ 1000 sec = 1.1 sec

So the answers A 1.1 sec is correct.