Respuesta :
Answer:
(a) [tex]1.59\times10^{-8}[/tex]
(b) [tex]3.30\times10^{-11}[/tex]
(c) [tex]1.87\times10^{-11}[/tex]
(d) [tex]4.58\times10^{-12}[/tex]
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:
[tex]{^{26} P_8} = \frac{26!}{(26-8)!}= 62990928000[/tex]
The probability of selecting the correct password if 8 different lower-case letters are used is:
[tex]P(Correct\ Password)=\frac{1000}{62990928000} = 1.59\times10^{-8}[/tex]
(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:
[tex]{^{52} P_8} = \frac{52!}{(52-8)!}= 30342338208000[/tex]
The probability of selecting the correct password if 8 different letters, some may be upper-case and it is case sensitive is:
[tex]P(Correct\ Password)=\frac{1000}{30342338208000} = 3.30\times10^{-11}[/tex]
(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,
[tex]P(Correct\ Password)=\frac{1000}{52^{8}} = 1.87\times10^{-11}[/tex]
(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:
[tex]P(Correct\ Password)=\frac{1000}{62^{8}} = 4.58\times10^{-12}[/tex]
