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

Answer:

(a) $1.59\times10^{-8}$

(b) $3.30\times10^{-11}$

(c) $1.87\times10^{-11}$

(d) $4.58\times10^{-12}$

Step-by-step 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:

${^{26} P_8} = \frac{26!}{(26-8)!}= 62990928000$

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

$P(Correct\ Password)=\frac{1000}{62990928000} = 1.59\times10^{-8}$

(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:

${^{52} P_8} = \frac{52!}{(52-8)!}= 30342338208000$

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

$P(Correct\ Password)=\frac{1000}{30342338208000} = 3.30\times10^{-11}$

(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,

$P(Correct\ Password)=\frac{1000}{52^{8}} = 1.87\times10^{-11}$

(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:

$P(Correct\ Password)=\frac{1000}{62^{8}} = 4.58\times10^{-12}$