/*! 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 4 The 鈥淪huffle鈥� button on Tami... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 4

The 鈥淪huffle鈥� button on Tamika鈥檚 CD player plays the songs in a random order. Tamika puts a four-song CD into the player and presses 鈥淪huffle.鈥� a. How many ways can the four songs be ordered? b. What is the probability that Song 1 will be played first? c. What is the probability that Song 1 will not be played first? d. Songs 2 and 3 are Tamika鈥檚 favorites. What is the probability that one of these two songs will be played first? e. What is the probability that Songs 2 and 3 will be the first two songs played (in either order)?

Short Answer

Expert verified
24 ways. Probability Song 1 first: 0.25. Probability Song 1 not first: 0.75. Probability Song 2 or 3 first: 0.5. Probability Songs 2 and 3 first two: 鈮� 0.167.

Step by step solution

01

Calculate Total Number of Orders (Part a)

The total number of ways to order four songs is given by the factorial of 4. Calculate: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]There are 24 ways to order the four songs.
02

Probability That Song 1 is Played First (Part b)

For Song 1 to be played first, we can arrange the remaining 3 songs in any order. Calculate the number of favorable outcomes: \[ 3! = 3 \times 2 \times 1 = 6 \]The probability is then the number of favorable outcomes divided by the total number of outcomes: \[ P(\text{Song 1 first}) = \frac{6}{24} = \frac{1}{4} = 0.25 \]
03

Probability That Song 1 is Not Played First (Part c)

The probability that Song 1 is not played first is the complement of the probability that Song 1 is played first. Calculate: \[ P(\text{Song 1 not first}) = 1 - P(\text{Song 1 first}) = 1 - 0.25 = 0.75 \]
04

Probability That Song 2 or Song 3 is Played First (Part d)

For Song 2 or Song 3 to be played first, we add the probabilities that Song 2 or Song 3 is played first. Each of these probabilities can be calculated similarly to Song 1. \[ P(\text{Song 2 or Song 3 first}) = P(\text{Song 2 first}) + P(\text{Song 3 first}) \]Knowing that each is \(\frac{1}{4}\): \[ P(\text{Song 2 or Song 3 first}) = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} = 0.5 \]
05

Probability That Songs 2 and 3 Are the First Two Songs (Part e)

First calculate the number of ways to order Songs 2 and 3 as the first two songs: \[ 2! = 2 \times 1 = 2 \]Then calculate the total number of ways to organize the remaining two songs: \[ 2! = 2 \times 1 = 2 \]The total number of favorable outcomes is: \[ 2 \times 2 = 4 \]The probability is then: \[ P(\text{Songs 2 and 3 first two}) = \frac{4}{24} = \frac{1}{6} \approx 0.167 \]

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.

Factorials
Factorials are a key concept in probability and combinatorics. A factorial, denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). This operation is essential in counting the number of ways to arrange a set of items. In our example exercise, we're arranging 4 songs. The total number of possible arrangements is found using \(4!\). The factorial function grows rapidly with larger numbers, so while it鈥檚 manageable with small numbers like 4, calculating larger factorials can result in very large outcomes.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations and permutations of objects. When considering the order of four songs, we use permutations, which are arrangements of a set where the order does matter. The exercise demonstrates this by asking how many ways the four songs can be ordered, which is \(4! = 24\). Another part of combinatorics evident here involves selecting specific outcomes, like determining how many ways Songs 2 and 3 can be the first two songs played, utilizing more complex combinations and permutations.
Independent Events
In probability, an event is deemed independent if the outcome of one event does not affect the outcome of another. In Tamika鈥檚 CD player shuffle scenario, each selection of a song to play first is an independent event. For instance, the choice of playing Song 1 first doesn鈥檛 influence the choice of playing Songs 2 or 3 next. This is important for calculating probabilities, as we assume each song has an equal chance of being selected at any sequence of the shuffle. Understanding independence is key in reflective thinking around such probability problems.
Probability Calculations
Probability calculations involve determining the likelihood of an event occurring. This is often represented as a fraction (successful outcomes over total possible outcomes), a decimal, or a percentage. In our problem set, the probability of Song 1 being played first is \(\frac{6}{24} \approx 0.25\), which means there is a 25% chance. Similarly, for combinations like having Songs 2 and 3 as the first two songs, understanding how to count each outcome precisely is critical and showcases the underlying use of both factorials and combinatorics in these calculations.
Favorable Outcomes
Favorable outcomes are the specific outcomes that satisfy the condition of the probability question. For example, the favorable outcome for Song 1 being played first is any possible arrangement where Song 1 is at the beginning. In the Tamika's CD example, out of the 24 permutations, 6 start with Song 1, thus making them favorable. When calculating the probability that either Songs 2 or 3 will be first, favorable outcomes include all permutations that start with either Song 2 or Song 3, totaling \(\frac{1}{2}\), or for both Songs 2 and 3 being first two in either order, we identify 4 such favorable outcomes from 24.

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

Automobile license plates in one state consist of three different letters followed by three different digits. The state does not use vowels or the letter \(Y\) , which prevents slang words from accidentally appearing on license plates. a. Each letter can appear only once on a given license plate. How many different sets of three letters are possible? b. Each digit can appear only once on a given license plate. How many different sets of three digits are possible? c. Altogether, how many different license plates with three letters followed by three digits are possible for this state?

In the Orange-and-White game, three white marbles and one orange marble are placed in a bag. A player randomly draws two marbles. If the marbles are different colors, the player wins a prize. a. List all the possible pairs in the sample space. (Hint: Label the marbles \(\mathrm{W} 1, \mathrm{W} 2\) \(\mathrm{W} 3,\) and \(\mathrm{O}\) .) b. What is the probability of winning a prize?

Ms. McDonald raises only chickens and pigs on her farm. If you know how many legs are in Ms. McDonald鈥檚 barn, you can find all the possible combinations of pigs and chickens. For example, if there are 6 legs, there could be 3 chickens, or 1 chicken and 1 pig. a. Copy and complete the table to show the possible combinations for different numbers of legs. The notation 3C-0P means 3 chickens and no pigs. b. Predict the number of combinations for 16 legs and for 18 legs. Check your predictions by listing all the possibilities. c. Challenge Write two expressions that describe the number of chicken-pig combinations for L legs. One of your expressions should be for L values that are multiples of 4; the other should be for L values that are not multiples of 4. d. There are 42 legs in the barn. Assuming each possible combination of pigs and chickens is equally likely, what is the probability that there are 8 pigs and 5 chickens in the barn?

Three whole numbers have a mean of 5. a. List all the whole-number triples with a mean of 5, and explain how you know you have found them all. b. How many such whole-number triples exist? c. Suppose all the whole-number triples with a mean of 5 are put into a hat, and one is drawn at random. What is the probability that at least two of the numbers in the triple are the same?

Imagine rolling five regular dice and looking for outcomes when all five dice match. a. How many different outcomes are possible on a roll of five dice? Explain. b. In how many of the possible outcomes do all five dice match? c. What is the probability of getting all five dice to match on a single roll? d. Suppose Tamika is given three rolls to get five matching dice. On the second and third rolls, she may roll some or all of the five dice again. On her first roll, Tamika gets three \(3 \mathrm{s}, \mathrm{a} 2\) and a \(6 .\) She picks up the dice showing 2 and 6 and rolls them again. What is the probability that she will get two more 3 \(\mathrm{s}\) on this roll?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.