/*! 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 72 Some three-digit numbers, such a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 72

Some three-digit numbers, such as 101 and \(313,\) read the same forward and backward. If you select a number from all threedigit numbers, find the probability that it will read the same forward and backward.

Short Answer

Expert verified
The probability of a randomly chosen three-digit number reading the same forward and backward (being a palindrome) is 0.1 or 10%.

Step by step solution

01

Finding Total Three-Digit Numbers

There are 900 total three-digit numbers, ranging from 100 to 999.
02

Finding Total Palindrome Numbers

In a palindromic three-digit number, the first (hundreds place) and the last digit (ones place) must be the same. This allows for 9 options (1-9, as 0 cannot be the first digit of a three-digit number) for the first number and the last digit simultaneously. The middle digit (tens place) can be any digit from 0 to 9, which gives 10 options. Therefore, the total palindrome three-digit numbers are \(9 \times 10 = 90\).
03

Calculate Probability

Probability is calculated as 'favourable outcomes' divided by 'total outcomes'. The favourable outcomes here are the 90 palindrome numbers and the total outcomes are the 900 total three-digit numbers. So, the required probability is \(\frac{90}{900} = \frac{1}{10} = 0.1\).

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.

Three-Digit Numbers
Three-digit numbers are numbers that range from 100 to 999. Each number consists of three digits placed in the hundreds, tens, and ones positions. This range gives us exactly 900 unique numbers. Here鈥檚 why: the smallest three-digit number is 100, and the largest is 999.
To find out how many numbers there are in this range, we simply subtract the smallest number (100) from the largest number (999) and add one to include the number 100 in our count.
  • The formula for this is: \[999 - 100 + 1 = 900\]
  • So, there are 900 three-digit numbers altogether.
Understanding this concept is essential when we calculate probabilities involving three-digit numbers.
Palindrome Numbers
A palindromic number is a number that reads the same forward and backward, meaning it mirrors itself. For three-digit numbers, the digits in the hundreds and ones place must be identical, while the digit in the tens place can vary freely.
To form a three-digit palindrome:
  • The hundreds and ones place must have the same digit. This digit can be any from 1 to 9, because a three-digit number cannot begin with 0. This gives us 9 possible choices.
  • The tens place, meanwhile, can contain any of the 10 digits from 0 to 9.
  • Therefore, the number of three-digit palindromic numbers can be calculated by multiplying the number of choices for the hundreds/ones place with the choices for the tens place: \[9 \times 10 = 90\]
Thus, there are 90 three-digit palindrome numbers.
Favourable Outcomes
In probability, an outcome is any possible result of an action. A favourable outcome is a result that fits the criteria we're interested in鈥攊n this case, a number reading the same forward as backward.
When calculating the probability that a randomly chosen three-digit number is a palindrome, we need to:
  • Identify the favourable outcomes. For this problem, they are the 90 palindromic numbers, as these meet our desired condition.
  • Determine the total possible outcomes, which are the total three-digit numbers, numbered from 100 to 999. We have already determined this as 900.
The formula for probability is: \[\text{Probability} = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\]Applying this to our situation gives us: \[\frac{90}{900} = \frac{1}{10} = 0.1\]Thus, the probability that a randomly selected three-digit number is a palindrome is 0.1 or 10%.

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

Use the formula for the value of an annuity to solve Exercises 77鈥�84. Round answers to the nearest dollar. At age \(25,\) to save for retirement, you decide to deposit \(\$ 75\) at the end of each month in an IRA that pays \(6.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age \(65 ?\) b. Find the interest.

Write an equation in point-slope form and slope-intercept form for the line passing through \((-2,-6)\) and perpendicular to the line whose equation is \(x-3 y+9=0 .\) (Section 2.4 Example \(2)\)

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(\\{H H, H T, T H, T T\\} .\) Find the probability of getting $$\text{the same outcome on each toss.}$$

A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January 10 how many degree-days are included from January 1 to January \(10 ?\)

You are taking a multiple-choice test that has eight questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.