Problem 56 - Applied Statistics and Probability for Engineers

Question

A computer system uses passwords that contain exactly eight characters, and each character is 1 of the 26 lowercase letters \((a-z)\) or 26 uppercase letters \((A-Z)\) or 10 integers \((0-9)\) Let \(\Omega\) 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 number of passwords in each of the following events. (a) \(\Omega\) (b) \(A\) (c) \(A^{\prime} \cap B^{\prime}\) (d) Passwords that contain at least 1 integer (e) Passwords that contain exactly 1 integer

Step-by-Step Solution

Answer

(a) \( 62^8 \), (b) \( 52^8 \), (c) \( 62^8 - 52^8 - 10^8 \), (d) \( 62^8 - 52^8 \), (e) \( 8 \times 10 \times 52^7 \)

Step 1: Calculate Total Possible Passwords (\( \Omega \))

To find the total number of passwords, note that there are 62 choices for each character (26 lowercase, 26 uppercase, 10 digits). Since passwords are exactly 8 characters, calculate \( 62^8 \):\[ \Omega = 62^8 \]

Step 2: Calculate Passwords with Only Letters (\( A \))

For passwords containing only letters, we have 52 choices for each character (26 lowercase + 26 uppercase). Calculate \( 52^8 \):\[ A = 52^8 \]

Step 3: Calculate Passwords with Only Integers (\( B \))

For passwords containing only integers, each character must be a digit, so there are 10 choices per character. Thus, calculate \( 10^8 \):\[ B = 10^8 \]

Step 4: Calculate Passwords with At Least One Letter and One Integer (\( A^{\prime} \cap B^{\prime} \))

First, calculate the passwords in \( A \cup B \) as those with only letters or only integers:\[ A \cup B = A + B = 52^8 + 10^8 \]Then, subtract from \( \Omega \) to find \( A^{\prime} \cap B^{\prime} \) (contains at least one letter and one digit):\[ A^{\prime} \cap B^{\prime} = \Omega - (A + B) \]

Step 5: Calculate Passwords with At Least One Integer

Passwords with at least one integer are all passwords minus those with only letters:\[ \text{At least one integer} = \Omega - A \]

Step 6: Calculate Passwords with Exactly One Integer

Choose exactly one position to be a digit (10 choices) and the remaining seven positions must be letters (52 choices each). There are 8 ways to choose which position is the digit:\[ \text{Exactly one integer} = \binom{8}{1} \times 10^1 \times 52^7 \]

Key Concepts

Password Combinations

When it comes to creating strong passwords, understanding the concept of password combinations is crucial for cybersecurity. In this context, a password is formed by combining characters from a defined set. For an 8-character password involving lowercase, uppercase letters, and digits, each position can have 26 lowercase letters, 26 uppercase letters, or 10 digits, totaling 62 possible choices per character. To find all possible 8-character combinations, you raise the number of choices (62) to the power of the password length (8). This gives us the total password space as \( \Omega = 62^8 \), which represents all potential passwords. Understanding this enormous combination number is essential for gauging password strength and security.

Event Probability

Event probability is about measuring how likely a certain subset of all possible outcomes, called an event, will occur. In password creation, events can be defined as passwords with specific characteristics, like those containing only letters or only digits. Each event has a probability determined by dividing the number of favorable outcomes by the number of total possible outcomes. For example, the event \( A \) represents passwords composed only of letters. If each character in these passwords can be one of 52 letters, the total number of such possible passwords is \( 52^8 \). The probability of this event happening, compared to all possible passwords, is \( \frac{52^8}{62^8} \). Knowing these probabilities assists in evaluating the likelihood of certain password types occurring and their momentum when it comes to different security policies.

Set Theory

Set theory is a foundational aspect of mathematics that deals with collections of objects, termed as sets. In the realm of password combinations, set theory helps us classify and handle events like passwords containing only letters or numbers. If \( \Omega \) is the set of all possible passwords, \( A \) is the set of passwords with only letters, and \( B \) is the set of passwords with only numbers, then the complement \( A' \) includes passwords not solely composed of letters and \( B' \) comprises those not purely made of numbers. To find hybrids, or passwords containing both letters and numbers, we look at \( A' \cap B' \), the intersection of the complements. This is computed as \( \Omega - (A + B) \), which subtracts purely lettered \( A \) and purely numbered \( B \) passwords from the total possibilities, leaving us with the mixed category.

Probability Calculations

Probability calculations in situations involving passwords often require combination logic and arithmetic. The straightforward problems involve computing basic probabilities, such as determining the probability that a randomly generated password has specific characteristics. For example, to calculate the probability of a password containing exactly one integer, you first decide which of the 8 positions the integer will occupy — there are 8 different spots. Within that spot, you have 10 numeric choices, leaving the remaining 7 positions for letters, with 52 choices each. The number of such specific passwords is calculated as \( \binom{8}{1} \cdot 10 \cdot 52^7 \). Dividing this by \( 62^8 \) provides the probability that a randomly chosen password will have exactly one integer. This process is critical for assessing risk or understanding the distribution of different password types in a set.