A computer system uses passwords that contain exactly eight characters, and each

Question

A computer system uses passwords that contain exactly eight characters, and each character is one of 26 lowercase letters (a-z) or 26 uppercase letters (A-Z) or 10 integers (0-9). Let X denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in X are equally likely. Determine the probability of each of the following:

(a) Password contains all lowercase letters given that it contains only letters
(b) Password contains at least 1 uppercase letter given that it contains only letters
(c) Password contains only even numbers given that it contains all numbers.

Explanation / Answer

Ans:

There are 62 permissible characters in a password, so there are
62^8 = 218,340,105,584,896 possible passwords in
because we are requiring all passwords to be 8 characters.

Of these, there are
52^8 = 53,459,728,531,456 passwords consisting only of letters (set A)
and
10^8 = 100,000,000 consisting only of integers (set B).


A) Probability that password contains all lowercase letters given that it contains only letters
= 26^8 / (62^8 - 10^8)
= 0.0009564

B)  Password contains at least 1 uppercase letter given that it contains only letters
=1-Probability that password contains all lowercase letters or no uppercase letters given that it contains only letters
=1-0.0009564

=0.9990

C) probability that password contains only even numbers given that it contains all numbers.

=5^8/10^8

=0.5^8

=0.0039

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.