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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.RS.2a
Chapter 3, Problem 3.RS.2a

You work in the security department of a bank’s website. To access their accounts, customers of the bank must create an 8-digit password. It is your job to determine the password requirements for these accounts. Security guidelines state that for the website to be secure, the probability that an 8-digit password is guessed on one try must be less than 1/60^8, assuming all passwords are equally likely.
Your job is to use the probability techniques you have learned in this chapter to decide what requirements a customer must meet when choosing a password, including what sets of characters are allowed, so that the website is secure according to the security guidelines.
2. Answering the Question
a. What password requirements would you set? What characters would be allowed?

Verified step by step guidance
1
Step 1: Understand the problem. The goal is to ensure that the probability of guessing an 8-digit password in one try is less than 1/(60^8). This involves calculating the total number of possible passwords and ensuring it is large enough to meet the security guideline.
Step 2: Define the total number of possible passwords. If the password consists of 8 characters, and each character can be chosen from a set of 'n' possible characters, the total number of possible passwords is given by n^8. Use the formula: n8.
Step 3: Set the inequality for security. To meet the security guideline, the total number of possible passwords must satisfy the inequality: n8 > 60^8. Solve this inequality to determine the minimum value of 'n', the size of the character set.
Step 4: Determine the character set. Based on the value of 'n' obtained in Step 3, decide what characters should be included in the password. For example, if 'n' is 62, you might include uppercase letters (26), lowercase letters (26), and digits (10). If 'n' is larger, you may need to include special characters.
Step 5: Formulate the password requirements. Based on the character set determined in Step 4, specify the password requirements. For example, you might require that passwords include at least one uppercase letter, one lowercase letter, one digit, and one special character, and be exactly 8 characters long.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it helps determine how likely it is for a randomly generated password to be guessed correctly. Understanding probability is crucial for assessing the security of password combinations and ensuring that the chance of unauthorized access remains low.
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Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In the context of password creation, it helps calculate the total number of possible passwords based on the allowed characters and their arrangement. This calculation is essential for determining whether the number of potential passwords meets the security guidelines set by the bank.

Character Sets

Character sets refer to the groups of characters that can be used to create passwords, such as lowercase letters, uppercase letters, numbers, and special symbols. The choice of character sets directly impacts the total number of possible passwords. By expanding the character set, the bank can significantly increase the complexity and security of the passwords, making them harder to guess.
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Related Practice
Textbook Question

In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.

22. Getting high grades and being awarded an academic scholarship

Textbook Question

A person's building access code is their first and last initials and four digits.

You know a person's first name only, and you know that the last digit is odd. What is the probability of guessing this person's code on the first try?

Textbook Question

4. The table on the left shows the secondary school student enrollment levels (in thousands by grade) in Oklahoma and Texas schools in a recent year. (Source: U.S. Nation

for Education Statistics)

A student in one of the indicated grades and states is randomly selected. Find the probability of selecting a student who

d. is enrolled in Texas, given that the student is in twelfth grade.

Textbook Question

You work in the security department of a bank’s website. To access their accounts, customers of the bank must create an 8-digit password. It is your job to determine the password requirements for these accounts. Security guidelines state that for the website to be secure, the probability that an 8-digit password is guessed on one try must be less than 1/60^8, assuming all passwords are equally likely.

Your job is to use the probability techniques you have learned in this chapter to decide what requirements a customer must meet when choosing a password, including what sets of characters are allowed, so that the website is secure according to the security guidelines.

3. For additional security, each customer creates a 5-digit PIN (personal identification number). The table on the right shows the 10 most commonly chosen 5-digit PINs. From the table, you can see that more than a third of all 5-digit PINs could be guessed by trying these 10 numbers. To discourage customers from using predictable PINs, you consider prohibiting PINs that use the same digit more than once.

b. Would you decide to prohibit PINs that use the same digit more than once? Explain.

Textbook Question

In Exercises 29-32, find the probability.

31. A 12-sided die, numbered 1 to 12, is rolled. Find the probability that the roll results in an odd number or a number less than 4.

Textbook Question

In Exercises 35–38, the bar graph shows the results of a survey in which 8806 undergraduate students were asked how many hours they spend on studying and other academic activities outside of class in a typical week. (Source: American College Health Association)

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37. Find the probability of randomly selecting an undergraduate who does not study from 6 to 10 hours per week.