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Homework answers / question archive / Assume you want to design an algorithm that auto-generates a 5 character password
Assume you want to design an algorithm that auto-generates a 5 character password. You determine that the algorithm will only choose from the following characters:
Remember that the algorithm will only choose from the characters listed above.
(a) Assume the algorithm chooses randomly from the entire list of possibilities listed above for each character (with the possibility of repeating a character). How many unique 5 character passwords are possible?
(b) Find the probability that the algorithm generates a 5 character password with only two integers where the first and the fifth characters are integers and are not the same. (i.e, the 5 character password begins and ends with two different integers and the other characters are non-integers)
(c) Find the probability that the algorithm generates a 5 character password with exactly two special symbols (with repeat characters allowed).
(d) Find the probability that at least one of the characters in the 5 characters password is either an upper or lowercase vowel.
(e) A hacker trying to break one of your 5 character passwords runs an algorithm that accurately identifies the first 2 characters of the password before being stopped. However, the hacker's algorithm has determined that the last 3 characters have exactly one special symbol, one lower case letter, and one integer. What is the probability the hacker successfully finds the correct password?
Answer:
(a) 9,765,625.
(b) 0.0311.
(c) 0.0205.
(d) 0.9222.
(e) 0.016.
Step-by-step explanation
The total no.of upper cases are 5, lower cases are 5, special symbols are 5, and integers are 10.
The no.of characters in the password is 5.
(a)
The no.of characters that can be used to make a password is 25, and the repetitions are allowed.
Each character in the password to fill with the possibility of 25 characters.
The total 5 characters in the password to fill with the possibility of 255 = 9,765,625.
Therefore, the no.of possible unique 5 character passwords is 9,765,625.
(b)
The no.of possibilities for the first character of the password to fill with any one of the numbers is 10.
The no.of possibilities for the last character of the password can be filled with any one of the numbers except the one number which is filled in the first character is 9.
The middle three characters to fill with any character other than numerics.
The no.of characters other than numerics is 15.
The number no.of possibilities to fill the middle three characters in 153 = 3,375 ways.
Therefore, the total no.of possibilities are 10 x 9 x 3,375 = 303,750.
The probability of the first and last characters with different numerics is:
(c)
The no.of ways two characters can be filled with 2 special symbols with repetitions is 5 x 5 = 25.
The remaining 3 characters in the password can be filled without special symbols.
The no.f characters without special symbols is 20.
The no.of possibilities to fill the remaining three characters are 20 x 20 x 20 = 8,000.
Therefore, the total no.of possibilities is 25 x 8,000 = 200,000
The probability of exactly two special symbols with repetitions is:
(d)
The no.of characters without either lowercase or uppercase vowels is 15.
The no.of ways to fill the 5 password characters without lowercase and uppercase vowels is 15 x 15 x 15 x 15 x 15 = 759,375.
The no.of ways to fill the 5 password characters with at least 1 of the characters either lowercase or uppercase is 9,765,625 - 759,375 = 9,006,250.
The probability of at least 1 of the 5 characters contains either lowercase or uppercase is:
(e)
Here the first 2 characters are identified correctly. Hence, the no.of ways for the first 2 characters is 1.
The next 3 characters have exactly 1 integer, 1 lowercase, and 1 special symbol.
The no.of ways to fill the last 3 characters in the password is 10 x 5 x 5 = 250.
The total no.of possible ways to fill is 250.
The total no.of possible to fill the last 3 characters without having any idea is 25 x 25 x 25 = 15,625.
The probability of the hacker finds the correct password is:
PFA
