Question

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

(a) P(A ∪ B)

A. {36^8/62^8}
B. {52^8/62^8}
C. {62^8/36^8}
D. {26^8/62^8}

Answer 1

The correct probability of a password being created with either all letters or all numbers is P(A ∪ B) = {52^8/62^8}, given that A represents passwords with only letters and B represents passwords with only integers.

Step-by-step explanation:

In the given computer system, the event A is the creation of passwords with only letters (either lowercase or uppercase), and the event B is the creation of passwords consisting only of integers. Since A and B are mutually exclusive events (a password cannot be made of only letters and only numbers at the same time), we can find the probability of the union of A and B, denoted as P(A ∪ B), by adding the probabilities of A and B together.

The total number of different passwords that can be created with 8 characters from the set of lowercase letters, uppercase letters, and integers (62 possible characters for each position) is 62^8. The total number of different passwords that consist of only letters (52 possible characters) is 52^8, and the total number of passwords that consist of only numbers (10 possible characters) is 10^8.

Using the addition rule of mutually exclusive events:

• P(A) = 52^8/62^8
• P(B) = 10^8/62^8
• P(A ∪ B) = P(A) + P(B) = 52^8/62^8 + 10^8/62^8

We compute P(A ∪ B) by adding the probabilities of A and B, and after simplification, we obtain P(A ∪ B) = (52^8 + 10^8)/62^8. Hence, the answer demonstrating the probability of creating a password with either all letters or all integers is option B. {52^8/62^8}.