91Ó°ÊÓ

A computer system uses passwords that are exactly seven characters and each character is one of the 26 letters \((a-z)\) or 10 integers \((0-9)\). You maintain a password for this computer system. Let \(A\) denote the subset of passwords that begin with a vowel (either \(a, e, i, o,\) or \(u\) ) and let \(B\) denote the subset of passwords that end with an even number (either \(0,2,\) \(4,6,\) or 8 ). (a) Suppose a hacker selects a password at random. What is the probability that your password is selected? (b) Suppose a hacker knows your password is in event \(A\) and selects a password at random from this subset. What is the probability that your password is selected? (c) Suppose a hacker knows your password is in \(A\) and \(B\) and selects a password at random from this subset. What is the probability that your password is selected?

Step by step solution

01

Calculate Total Number of Passwords

Each password is exactly 7 characters long, and each character can be one of 36 possibilities (26 letters + 10 numbers). Hence, the total number of possible passwords is given by raising the total number of options for one character to the power of the length of the password: \(36^7\).

02

Calculate Passwords in Subset A

Subset \(A\) consists of passwords starting with a vowel. There are 5 vowels and 36 options for each of the remaining 6 places, so the number of passwords starting with a vowel is \(5 \times 36^6\).

03

Calculate Passwords in Subset B

Subset \(B\) consists of passwords ending with an even number. There are 5 even numbers and 36 options for each of the first 6 places, so the number of passwords ending with an even number is \(36^6 \times 5\).

04

Calculate Passwords in Subset A and B

Subset \(A \cap B\) includes passwords that both start with a vowel and end with an even number. This gives \(5\) choices for the first character, 36 choices each for the 5 middle characters, and \(5\) choices for the last character. Hence, the number of passwords in \(A \cap B\) is \(5 \times 36^5 \times 5\).

05

Calculate Probability for Question (a)

The probability that the hacker selects your specific password is \(\frac{1}{36^7}\) since any single password is equally likely to be selected out of the total possible.

06

Calculate Probability for Question (b)

Given that the hacker selects from event \(A\), the probability that your specific password is selected from \(A\) is \(\frac{1}{5 \times 36^6}\).

07

Calculate Probability for Question (c)

Given the hacker selects from event \(A \cap B\), the probability that your specific password is selected from this subset is \(\frac{1}{5 \times 36^5 \times 5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. In the context of password security, combinatorics helps us calculate the number of possible passwords. To find the total number of possible passwords, we consider all the choices for each position in the password.

• Each password is seven characters long.
• Each character can be one of the 26 letters or 10 numbers, totaling 36 choices per character.
• The total number of combinations is determined by taking these choices for each position, leading to \(36^7\) total passwords.

Understanding these basic principles allows us to compute how many specific types of passwords exist, such as those starting with a vowel or ending with a digit. By breaking down the problem into manageable parts, we can leverage combinatorial principles to handle complex password-related scenarios.

Event Subsets

Event subsets involve the idea of choosing a specific portion of a set based on certain criteria. For passwords, subsets are groups of passwords that share common characteristics. In the exercise, we have two important subsets:

• Subset A: Passwords that start with a vowel. There are 5 vowels (a, e, i, o, u), and for every vowel choice, the remaining 6 characters have 36 options each.
• Subset B: Passwords ending with an even number. The choices work similarly, with 5 possible even numbers and 36 options for the first 6 characters.

To combine these subsets, we look at \(A \cap B\), which is the intersection where passwords meet both criteria: they start with a vowel and end with an even number. This requires a combinatorial calculation where we fix the first and last character and consider the possibilities for the middle characters. By understanding event subsets, we can better grasp the probability scenarios presented in the exercise.

Password Security

Password security is crucial in protecting digital systems. One aspect that makes passwords secure is the sheer number of possible combinations available to a potential hacker. The larger the number of potential passwords, the harder it is for a hacker to guess one correctly.

• Using a mix of letters and numbers increases complexity.
• Longer passwords are generally more secure due to the exponential increase in possibilities; for instance, 36 choices over 7 slots results in an enormous \(36^7\) combinations.

Security in passwords is enhanced by using subsets of event criteria to enforce rules like starting with a vowel or ending with an even number. Each added constraint reduces the total number but also adds a layer of specificity, which can either hinder or help in security depending on known information about the password. This exercise teaches us how combinatorics and event subsets contribute to the security measures one might use in real-world applications.