/*! 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 6 You pick an integer at random be... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 6

You pick an integer at random between zero and \(10^{5}\) inclusive. What is the probability that its digits are all different?

Short Answer

Expert verified
The probability is approximately 0.1686.

Step by step solution

01

Determine Total Possibilities

An integer between 0 and \(10^5\) inclusive ranges from 0 to 100000. Therefore, there are \(100,001\) possible integers.
02

Identify the Valid Range of Integers

Integers with all distinct digits can have up to 6 digits, since any number with more than 6 digits would repeat a digit due to having only 10 digits (0-9).
03

Calculate Possibilities for Each Digit Length

1-Digit: There are 10 such numbers (0-9).2-Digits: Choose any non-zero digit (1-9) for the first digit and any remaining digit (0-9 without the first digit) for the second. That gives \(9 \times 9 = 81\) choices.3-Digits: Choose the first digit (1-9), second digit avoiding the first, third digit avoiding the first two; resulting in \(9 \times 9 \times 8 = 648\).4-Digits: Choose four digits avoiding previous choices giving \(9 \times 9 \times 8 \times 7 = 4536\).5-Digits: Choose five digits in sequence avoiding repeats results in \(9 \times 9 \times 8 \times 7 \times 6 = 27216\).6-Digits: Choose the first digit out of 1-9, then the remaining 9 digits in decreasing order: \(9 \times 9 \times 8 \times 7 \times 6 \times 5 = 136080\).
04

Sum All Valid Possibilities

Adding all the valid possibilities calculated for each digit length: \[10 + 81 + 648 + 4536 + 27216 + 136080 = 168571\]
05

Calculate the Probability

The probability is the quotient of the total number of integers with distinct digits over the total possible integers: \[ \frac{168571}{100001} \approx 0.1686 \]

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.

Combinatorics
Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It helps us determine the number of ways we can choose or arrange elements according to certain rules or conditions. When trying to solve problems related to counting probabilities, like the one provided, combinatorics is the fundamental tool that enables us to calculate possibilities systematically.
In the context of this exercise, combinatorics helps us determine how many numbers can be formed with distinct digits. We use it to calculate the possible combinations of digits for numbers of varying lengths. By considering each place value and the choices available for distinct digits, we methodically work through one-digit up to six-digit numbers. This logical progression ensures all distinct possible combinations are considered.
Distinct Digits
When we talk about distinct digits, we are referring to numbers that do not repeat any digit. For example, 123 is a number with distinct digits, but 112 is not. Distinct digits are essential in solving the exercise, as it asks for the probability of randomly selecting an integer with all unique digits from a given range.
To find such numbers, we imagine each position in a number being filled with different digits from the set {0, 1, 2, ..., 9}. The challenge is to ensure no digit is repeated. This exercise shows how to approach finding numbers of different lengths (from one-digit to six-digits) and successfully choosing digits for these places while avoiding repetition. Learning how to effectively determine distinct sets of digits is crucial in understanding the broader concept of permutations, which are arrangements of objects or numbers.
Integer Range
The integer range is the set of whole numbers that we are considering for the exercise. Given our range from 0 to 100,000 inclusive, the task is to explore this spectrum to understand how many numbers have distinct digits. The total integer range is calculated first, which is given as 100,001 possible integers.
The relevance of the integer range here is the need to calculate probabilities accurately. Knowing the full range allows us to compare the subset of numbers with distinct digits to the entire set, facilitating the probability calculation. Our integer range in this problem serves as the denominator when we ultimately calculate the likelihood of picking an integer with distinct digits, impacting the accuracy of our answer.
Probability Calculation
Probability calculation is crucial in understanding the likelihood of an event occurring. In our exercise, we want to know the probability that a randomly chosen integer from our specified range has distinct digits. We start with a simple concept: probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.
  • Total possibilities: 100,001 integers (we include zero, noting the range is inclusive).
  • Favorable outcomes: Numbers with distinct digits calculated through combinatorial methods.
To arrive at the solution, the favorable outcomes (168,571 integers with distinct digits) are divided by the total possibilities (100,001). This calculation gives the probability, approximating it to 0.1686. Calculating probabilities helps us predict behaviors or outcomes and is widely applicable in real-life scenarios such as risk analysis, finance, and decision-making situations.

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

An orienteer runs on the rectangular grid through the grid points \((m, n), m, n=0,1,2, \ldots\) of a Cartesian plane. On reaching \((m, n)\), the orienteer must next proceed either to \((m+1, n)\) or \((m, n+1)\). (a) Show the number of different paths from \((0,0)\) to \((n, n)\) equals the number from \((1,0)\) to \((n+1, n)\) and that this equals \(\left(\begin{array}{c}2 n \\ n\end{array}\right)\), where \(\left(\begin{array}{l}k \\ r\end{array}\right)=\frac{k !}{r !(k-r) !}\). (b) Show that the number of different paths from \((1,0)\) to \((n+1, n)\) passing through at least one of the grid points \((r, r)\) with \(1 \leq r \leq n\) is equal to the total number of different paths from \((0,1)\) to \((n+1, n)\) and that this equals \(\left(\begin{array}{c}2 n \\ n-1\end{array}\right)\). (c) Suppose that at each grid point the orienteer is equally likely to choose to go to either of the two possible next grid points. Let \(A_{k}\) be the event that the first of the grid points \((r, r), r \geq 1\), to be visited is \((k, k)\). Show that $$ \mathbf{P}\left(A_{k}\right)=\frac{4^{-k}}{2 k-1}\left(\begin{array}{c} 2 k-1 \\ k \end{array}\right) $$

A bag contains \(b\) black balls and \(w\) white balls. If balls are drawn from the bag without replacement, what is the probability \(P_{k}\) that exactly \(k\) black balls are drawn before the first white ball? By considering \(\sum_{k=0}^{b} P_{k}\), or otherwise, prove the identity $$ \sum_{k=0}^{b}\left(\begin{array}{l} b \\ k \end{array}\right) /\left(\begin{array}{c} b+w-1 \\ k \end{array}\right)=\frac{b+w}{w} $$ for positive integers \(b, w\).

The points \(A_{0}, A_{1}, \ldots, A_{n}\) lie, in that order, on a circle. Let \(a_{1}=1, a_{2}=1\) and for \(n>2\), let \(a_{n}\) denote the number of dissections of the polygon \(A_{0} A_{1} \ldots A_{n}\) into triangles by a set of noncrossing diagonals, \(A_{i} A_{j}\). (a) Check that \(a_{3}=2\) and \(a_{4}=5\). (b) Show that in each dissection there is a unique \(i(1 \leq i \leq n-1)\) such that cuts are made along both \(A_{0} A_{i}\) and \(A_{n} A_{i}\). (c) Show that \(a_{n}=a_{1} a_{n-1}+a_{2} a_{n-2}+\cdots+a_{n-2} a_{2}+a_{n-1} a_{1}\). (d) If \(f(x)=\sum_{1}^{\infty} a_{n} x^{n}\), show (by considering the coefficient of each power of \(x\) ) that \((f(x))^{2}-\) \(f(x)+x=0\), and show that \(\left.f(x)=\frac{1}{2}-\frac{1}{2} \sqrt{(} 1-4 x\right)\).

An urn contains \(b\) blue balls and \(a\) aquamarine balls. The balls are removed successively at random from the urn without replacement. If \(b>a\), show that the probability that at stages until the urn is empty there are more blue than aquamarine balls in the urn is \((b-a) /(a+b)\). Why is this result called the ballot theorem?

The probability that an archer hits the target when it is windy is \(0.4\); when it is not windy, her probability of hitting the target is \(0.7\). On any shot, the probability of a gust of wind is \(0.3\). Find the probability that: (a) On a given shot, there is a gust of wind and she hits the target. (b) She hits the target with her first shot. (c) She hits the target exactly once in two shots. (d) There was no gust of wind on an occasion when she missed.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Mechanics 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.