Problem 35 35\. Four candidates are running... [FREE SOLUTION] (2024)

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Chapter 16: Problem 35

35\. Four candidates are running for mayor of Happyville. According to thepolls candidate $A$ has a "one in five" probability of winning [i.e.,$\operatorname{Pr}(A)=1 / 5] .$ Of the other three candidates, all we know isthat candidate $C$ is twice as likely to win as candidate $B$ and thatcandidate $D$ is three times as likely to win as candidate $B$. Find theprobability assignment for this probability space.

Short Answer

Expert verified

The probabilities of each candidate winning are as follows: A: 1/5, B: 2/15, C: 4/15, D: 2/5.

Step by step solution

Introduction

Let $B$, $C$, and $D$ represent the probabilities of candidates B, C, and D winning respectively. It's given that $A$ has a probability of $1/5$ to win and $C$ is twice as likely as $B$ to do so, while $D$ is three times likely as $B$. So, we can say that $C = 2B$ and $D = 3B$. The sum of all probabilities must equal 1, hence we formulate the equation $1/5 + B + 2B + 3B = 1$.

Solve the equation

To find the values of $B$, $C$, and $D$, solve the equation. Simplify by combining similar terms to form $6B + 1/5 = 1$. Then, subtract $1/5$ from both sides of the equation to isolate $6B$ on one side and get $6B = 4/5$. Divide both sides by 6 to solve for $B$. Thus, $B = (4/5) / 6$, or simplified, $B = 2/15$.

Calculate the probabilities for C and D

Now calculate the probabilities for $C$ and $D$ using the values for $B$ determined in step 2 and the relationships given in the problem. As mentioned above, $C = 2B$ and $D = 3B$. Therefore, $C = 2 * (2/15) = 4/15$, and $D = 3 * (2/15) = 6/15$, which simplifies to $D = 2/5$.

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Most popular questions from this chapter

Two teams (call them $X$ and $Y$ ) play against each other in the WorldSeries. (The World Series is a best-of-seven series: The first team to winfour games wins the series, and games cannot end in a tic.) We can describe anoutcome for the World Series by writing a string of letters that indicate (inorder) the winner of each game. For example, the string $X Y X X Y X$represents the outcome: $X$ wins game $1, Y$ wins game $2, X$ wins game $3,$and so on. (a) Describe the event $" X$ wins in five games" (b) Describe the event "the series is over in game $5 . "$ (c) Describe the event "the series is over in game 6 " (d) Find the size of the sample space.A pair of dice is rolled. The observation is the number that comes up on eachdie (see Example 16.4 ). Write out the event described by each of thefollowing statements as a set. (a) $E_{1}$ " pairs are rolled." (A "pair" is a roll in which both dice comeup the same number.) (b) $E_{2}:$ "craps are rolled." ("Craps" is a roll in which the sum of thetwo numbers rolled is $2,3,$ or $12 .)$ (c) $E_{3}:$ "a natural is rolled." (A "natural" is a roll in which the sum ofthe two numbers rolled is 7 or $11 .$ )The Brute Force Bandits is a punk rock band planning their next concert tour.The band has a total of 30 new songs in their repertoire. (a) How many different set lists of 18 songs can the band select to play onthe tour? (Hint: Assume that the order in which the songs are listed isirrelevant.) (b) How many different ways are there for the band to record a CD consistingof 18 songs chosen from the 30 new songs? (Hint: In recording a CD, the orderin which the songs appear on the $\mathrm{CD}$ is relevant.)Suppose that you roll a single die. If an odd number (1,3,0 5) comes up, you win the amount of your roll (\$1, \$3, or \$5 respectively).If an even number $(2,4,$ or 6$)$ comes up, you have to pay the house theamount of your roll $(\$ 2, \$ 4,$ or $\$ 6$ respectively). (a) Find the expected payoff for this game. (b) Is this a fair game? Explain.A fair coin is tossed three times. Find the expected number of heads that comeup.

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Problem 35 35\. Four candidates are running... [FREE SOLUTION] (2024)

Short Answer

Step by step solution

Introduction

Solve the equation

Calculate the probabilities for C and D

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Applied Mathematics

Mechanics Maths

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Privacy Overview

References