/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 90 Common names The Census Bureau s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 90

Common names The Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Brown, Jones, Miller, Davis, Garcia, Rodriguez, and Wilson. These names account for 9.6% of all U.S. residents. Out of curiosity, you look at the authors of the textbooks for your current courses. There are 9 authors in all. Would you be surprised if none of the names of these authors were among the 10 most common? (Assume that authors鈥� names are independent and follow the same probability distribution as the names of all residents.)

Short Answer

Expert verified
There's about a 46.3% chance that none of the authors have a common name. This is a reasonable likelihood, so it's not surprising.

Step by step solution

01

Understanding Probability

We want to calculate the probability that none of the authors' names appear in the top 10 most common names. According to the problem, 9.6% of U.S. residents have one of these common names. Therefore, the probability that a single person does not have one of these common names is 1 - 0.096 = 0.904.
02

Event Probability for One Author

For each author, the probability that their name does not appear among the 10 most common names is 0.904, as calculated earlier. Thus, if we have one author, the probability of this author not having a common name is 0.904.
03

Combined Probability for All Authors

We now have 9 authors, each with a probability of 0.904 of not having a common name. Since the authors are independent, the combined probability is the product of individual probabilities. Therefore, the probability that none of the 9 authors have a common name is calculated as follows: \(P = (0.904)^9\).
04

Calculate the Final Probability

Calculate \((0.904)^9\): \(P = 0.904^9 \approx 0.463\). This result tells us that there is approximately a 46.3% chance that none of the authors have one of the common names.
05

Interpretation of Result

Since there is around a 46.3% probability of none of the authors having one of the top 10 common names, this suggests a reasonable likelihood that this event could occur. Therefore, it would not be particularly surprising if none of the author's names were among those common names.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Independent Events
In probability theory, when we talk about independent events, we mean events where the occurrence of one event does not affect the occurrence of another. This is a key principle when analyzing situations involving multiple events.
For our given problem, the probability that one author does not have a common surname is 0.904. This likelihood is independent of the names of the other authors. Thus, you're looking at 9 independent probabilities that each author does not share a surname with the most common names list.
  • This independence means we can multiply the probabilities of each author鈥檚 name not being common to find the overall probability.
  • This is a practical aspect in probability as it allows simplifying complex scenarios.
Without the concept of independent events, calculating how often a set of names doesn't show commonality would be much harder.
Probability Distribution
A probability distribution provides a mathematical description of the possible outcomes of a random variable. Each possible outcome has a probability associated with it, detailing how likely it is to occur.
In this exercise, the underlying probability distribution is simple due to the commonality of certain surnames in the population. With 9.6% of the population sporting one of the top 10 names, this acts as the foundation for calculating how frequently these names appear among any group of people.
  • The event space here is binary for each author: either they have one of the 10 common surnames or they do not.
  • The probability of having a common name is 0.096, while not having one is 0.904.
Understanding this distribution concept is vital to setting up the subsequent calculations accurately.
Probability Calculation
The core of our exercise lies in calculating a specific probability: that none of the 9 authors share a top 10 common surname. Calculating such probabilities involves breaking down the problem using basic probability rules.
If the probability that one author does not have a common surname is 0.904, then for 9 authors, the combined probability that none have a common surname involves repeated multiplication. This leads us to the formula: \[(0.904)^9\]
  • Exponential power helps compactly represent repeated events with the same probability.
  • It emphasizes the impact of multiple independent trials on overall likelihood.
This calculation method thoroughly highlights the power and utility of understanding and applying probability calculations to solve real-world problems.
Common Names Statistics
Statistics give us a snapshot of how common certain names are within a population. The Census Bureau provides such figures for the United States, indicating which names dominate in frequency.
Understanding these statistics is crucial as they provide tangible data we can use in probability calculations. In our example, knowing that 9.6% of people have one of the most common surnames is essential to set up accurate probability models.
  • This statistic directly impacts how often we expect to encounter these common names in daily life and specific groups.
  • This forms the basis of assumptions like those made in the problem: that probabilities for individual names align with broader population statistics.
Data-driven insights from common names statistics play a pivotal role in making informed predictions about such probabilities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Mac or PC? A recent census at a major university revealed that 40% of its students primarily used Macintosh computers (Macs). The rest mainly used PCs. At the time of the census, 67% of the school鈥檚 students were undergraduates. The rest were graduate students. In the census, 23% of respondents were graduate students who said that they used PCs as their main computers. Suppose we select a student at random from among those who were part of the census. (a) Assuming that there were 10,000 students in the census, make a two-way table for this chance process. (b) Construct a Venn diagram to represent this setting. (c) Consider the event that the randomly selected student is a graduate student who uses a Mac. Write this event in symbolic form using the two events of interest that you chose in (b). (d) Find the probability of the event described in (c). Explain your method.

Education among young adults Choose a young adult (aged 25 to 29) at random. The probability is 0.13 that the person chosen did not complete high school, 0.29 that the person has a high school diploma but no further education, and 0.30 that the person has at least a bachelor鈥檚 degree. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor鈥檚 degree? Why? (b) What is the probability that a randomly chosen young adult has at least a high school education? Which rule of probability did you use to find the answer?

The geometric distributions You are tossing a pair of fair, six-sided dice in a board game. Tosses are independent. You land in a danger zone that requires you to roll doubles (both faces showing the same number of spots) before you are allowed to play again. How long will you wait to play again? (a) What is the probability of rolling doubles on a single toss of the dice? (If you need review, the possible outcomes appear in Figure 5.2 (page 300). All 36 outcomes are equally likely.) (b) What is the probability that you do not roll doubles on the first toss, but you do on the second toss? (c) What is the probability that the first two tosses are not doubles and the third toss is doubles? This is the probability that the first doubles occurs on the third toss. (d) Now you see the pattern. What is the probability that the first doubles occurs on the fourth toss? On the fifth toss? Give the general result: what is the probability that the first doubles occurs on the kth toss?

Mac or PC? A recent census at a major university revealed that 40% of its students mainly used Macintosh computers (Macs). The rest mainly used PCs. At the time of the census, 67% of the school鈥檚 students were undergraduates. The rest were graduate students. In the census, 23% of the respondents were graduate students who said that they used PCs as their primary computers. Suppose we select a student at random from among those who were part of the census and learn that the student mainly uses a Mac. Find the probability that this person is a graduate student. Show your work.

Late shows Some TV shows begin after their scheduled times when earlier programs run late. According to a network鈥檚 records, about 3% of its shows start late. To find the probability that three consecutive shows on this network start on time, can we multiply (0.97)(0.97)(0.97)? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.