Question

A hard-headed hacker is attempting to login into Larry’s Amazon account by guessing his password. The hacker does not give up until she tries 1 thousand different passwords. What is the probability that she will guess the password and break in if by rules, the password must consist of: a. 8 different lower-case letters

b. 8 different letters, some may be upper-case, and it is case sensitive
c. any 8 letters, upper- or lower-case, and it is case sensitive
d. any 8 characters including letters and digits

Answer 1

(a)
`1.59*10^(-8)`

(b)
`3.30*10^(-11)`

(c)
`1.87*10^(-11)`

(d)
`4.58*10^(-12)`

Explanation:

The password is of 8 characters.

The hacker tries 1000 times to get the correct password.

(a)

There are 26 letters in the English Alphabet series.

The number of ways to create a password of 8 different lower-case letters are:

`{^(26) P_8} = (26!)/((26-8)!)= 62990928000`

The probability of selecting the correct password if 8 different lower-case letters are used is:

`P(Correct\ Password)=(1000)/(62990928000) = 1.59*10^(-8)`

(b)

If both upper case and lower case letters are used, then there are 52 letters available to form a 8 characters password.

The number of ways to create a password of 8 different letters, some may be upper-case, and it is case sensitive is:

`{^(52) P_8} = (52!)/((52-8)!)= 30342338208000`

The probability of selecting the correct password if 8 different letters, some may be upper-case and it is case sensitive is:

`P(Correct\ Password)=(1000)/(30342338208000) = 3.30*10^(-11)`

(c)

The number of ways to form a 8 characters password using any 8 letters, upper- or lower-case, and it is case sensitive is, 52⁸.

Then the probability of selecting the correct password if any 8 letters, upper- or lower-case, and it is case sensitive is,

`P(Correct\ Password)=(1000)/(52^(8)) = 1.87*10^(-11)`

(d)

There are 10 digits (0 - 9), 26 uppercase letters and 26 lowercase letters.

Total available characters = 62.

The number of ways to form a 8 characters password using any 8 letters, upper- or lower-case or digits is = 62⁸

Then the probability of selecting the correct password if any 8 characters including letters and digits are used is:

`P(Correct\ Password)=(1000)/(62^(8)) = 4.58*10^(-12)`