Problem 141 - Applied Statistics and Probability for Engineers

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 \((0-9)\). Let \(\Omega\) denote the set of all possible password, and let \(A\) and \(B\) denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in \(\Omega\) are equally likely. Determine the following robabilities: (a) \(P\left(A \mid B^{\prime}\right)\) (b) \(P\left(A^{\prime} \cap B\right)\) (c) \(P\) (password contains exactly 2 integers given that it contains at least 1 integer)

Step-by-Step Solution

Answer

(a) \(P(A | B') = \frac{52^8}{62^8 - 10^8}\) (b) \(P(A' \cap B) = \frac{10^8}{62^8}\) (c) \(P\text{(exactly 2 integers | at least 1 integer)} = \frac{\binom{8}{2} \cdot 10^2 \cdot 52^6}{62^8 - 52^8}\)

Step 1: Determine Total Number of Passwords

The total number of possible characters for each position in the password is 62 (26 lowercase letters + 26 uppercase letters + 10 integers). Since the password is 8 characters long, the total number of possible passwords is \(|\Omega| = 62^8\).

Step 2: Calculate Number of Passwords Containing Only Letters

For passwords containing only letters (event \(A\)), each of the 8 positions can be filled with any of the 52 letters (26 lowercase + 26 uppercase). Hence, the number of such passwords is \(|A| = 52^8\).

Step 3: Calculate Number of Passwords Containing Only Integers

For passwords containing only integers (event \(B\)), each of the 8 positions can be filled with any of the 10 numbers. Hence, the number of such passwords is \(|B| = 10^8\).

Step 4: Find \(P(A \mid B^{\prime})\)

Since \(B'\) represents passwords that do not contain only integers, \(P(A \mid B')\) is calculated by considering the total set minus those in \(B\). The probability is given by \(P(A \mid B') = \frac{|A \cap B'|}{|B'|} = \frac{|A|}{|\Omega| - |B|}\).

Step 5: Calculate Probability of \(A' \cap B\)

\(A' \cap B\) are passwords that only contain integers, which was already calculated as \(|B| = 10^8\). So, \(P(A' \cap B) = \frac{|B|}{|\Omega|}\).

Step 6: Determine Probability of Password with Exactly 2 Integers Given At Least 1 Integer

Firstly, calculate \(P(E_2)\), the probability that a password contains exactly 2 integers: Choose 2 positions from 8 for integers (\(\binom{8}{2}\) ways), and fill those with integers (\(10^2\)) and the remaining 6 positions with letters (\(52^6\)). Then, divide by the total number of passwords. Calculate \(|F|\), the number of passwords containing at least 1 integer: \(|F| = |\Omega| - |A|\). Finally, \(P(E_2 \mid F) = \frac{|E_2|}{|F|}\).

Key Concepts

Combinatorics

Combinatorics is the field of mathematics that deals with counting, arrangement, and combination of objects. When attempting to solve password-related problems, combinatorics helps us calculate the total number of possible outcomes.
For example, in this exercise, we focus on passwords of 8 characters, where each character can be a lowercase letter, uppercase letter, or a digit.

• The total number of possibilities for each character is 62 (26 lowercase + 26 uppercase + 10 digits).
• To find the total number of possible passwords, we use the formula for permutations, since the order matters: \(|\Omega| = 62^8\).

Combinatorics also helps calculate specific events, like passwords only containing letters or only numbers. We use the formula for permutations over the restricted sets, for example:

• For passwords with only letters, calculate \(|A| = 52^8\), where 52 represents the total letters available.
• For passwords with only integers, calculate \(|B| = 10^8\), where 10 represents the total digits available.

Conditional Probability

Conditional probability is used to calculate the likelihood of an event occurring given that another event has already occurred. This is particularly useful when we deal with subsets of a larger probability space.
In our password problem, we are interested in situations like determining the probability of a password containing only letters given it does not solely contain numbers.
To solve these, we utilize the formula for conditional probability:

• \(P(A \mid B') = \frac{|A \cap B'|}{|B'|}\), which simplifies to \(\frac{|A|}{|\Omega| - |B|}\) because \(A\) doesn't overlap with \(B\).

Conditional probability also applies when determining the likelihood of a password having exactly two integers, given it contains at least one. We find the specific probability using:

• Identify passwords with exactly 2 integers: Choose 2 positions out of 8 for digits, fill with integers, and the rest with letters.
• Calculate the possibility of having at least 1 integer: \(|F| = |\Omega| - |A|\).
• Use \(P(E_2 \mid F) = \frac{|E_2|}{|F|}\).

Discrete Mathematics

Discrete mathematics encompasses a set of mathematical topics that involve countable, distinct elements. This field heavily influences probability theory, where we deal with finite sets and their elements.
In the context of our password problem, the set of all possible passwords \(\Omega\) and various subsets (like those containing only letters or only integers) display characteristics of discrete mathematics.

• The Cartesian product and set theory allow us to construct and analyze sets like \(A\) and \(B\), involving different character combinations.
• Using basic principles, we calculate outcomes and associated probabilities using formulas from discrete math. For instance, finding \(|F| = |\Omega| - |A|\) involves straightforward set subtraction due to discrete elements (passwords without any letters).

This discrete method helps us handle infinite-looking possibilities by treating them as finite, discrete sets with clear boundaries and rules. Thus, by applying these concepts, we unravel complex mathematical problems into manageable parts.