Question

A computer system uses passwords that contain exactly 4 characters, and each character is 1 of the 3 lowercase letters (a, b, c) or 3 upper case letters (A, B, C) or the 5 odd digits (1, 3, 5, 7, 9). Let Ω 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. Determine the probability that a password contains only numbers given that it contains no lower case letters.

Answer 1

total number of choices =N(11 choices for each charecter )=11⁴

P(no lower case letters) =N(for each of 4 charecters there are total 8 choices (3 upper case and 5 digits)

=(8⁴/11⁴)

and P(only numbers ) =(5⁴/11⁴)

therefore P(only numbers |no lower case letters) =P(only numbers )/P(no lower case letters)

=(5⁴/11⁴)/(8⁴/11⁴) =(5⁴/8⁴)=625/4096