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

Determine the number of outcomes $N$ in each sample spac (a) A student randomly answers a 10-question true-fals quiz. The student'sanswer $(T$ or $F)$ to each question observed. (b) A student randomly answers a 10-question multipl choice quiz. Thestudent's answer $(A, B, C, D,$ or $E)$ each question is observed.A coin is tossed 10 times in a row. The observation is how the coin lands ($H$ or $T$ ) on each toss (see Exercise 7 ). Write out the event described byeach of the following statements as a set. (a) $E_{1}:$ "none of the tosses comes up tails." (b) $E_{2}:$ "exactly one of the 10 tosses comes up tails" (c) $E_{3}$ : "nine or more of the tosses come up tails."Find the odds of each of the following events. (a) an event $E$ with $\operatorname{Pr}(E)=4 / 7$ (b) an event $E$ with $\operatorname{Pr}(E)=0.6$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.Joe is buying a new plasma TV at Circuit Town. The salesman offers Joe athree-year extended warranty for $\$ 80 .$ The salesman tells Joe that $24 \%$of these plasma TVs require repairs within the first three years, and theaverage cost of a repair is \$400. Should Joe buy the extended warranty?Explain your reasoning.

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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

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Logic and Functions

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Privacy Overview

References