/*! 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 36 Seven horses \((A, B, C, D, E, F... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 36

Seven horses \((A, B, C, D, E, F,\) and \(G)\) are running in the Boonsville Sweepstakes. According to the oddsmakers, \(A\) has a "one in four" probability of winning [i.e., \(\operatorname{Pr}(A)=1 / 4]\), \(B\) has a "three in ten" probability of winning, and \(C\) has a "one in twenty" probability of winning. The remaining four horses all have the same probability of winning. Find the probability assignment for the probability space.

Short Answer

Expert verified
The probability of each horse winning is as follows: \(\operatorname{Pr}(A)=1 / 4, \operatorname{Pr}(B)=3 / 10, \operatorname{Pr}(C)=1 / 20\), and for horses D, E, F, G it's determined by dividing the remaining probability space (i.e., \(1 - \operatorname{Pr}(A) - \operatorname{Pr}(B) - \operatorname{Pr}(C)\)) by 4.

Step by step solution

01

Calculate the added probabilities of horses A, B and C

From the problem, we have following probabilities: \(\operatorname{Pr}(A)=1 / 4, \operatorname{Pr}(B)=3 / 10,\) and \(\operatorname{Pr}(C)=1 / 20\). Now, find their sum.
02

Calculate the remaining probability

Since the probability for all the possible outcomes (i.e., any horse winning the race) should equal to 1, subtract the sum from Step 1 from 1. This will give the total probability that one of the horses D, E, F or G wins.
03

Determine the probability for each of horses D, E, F, G to win

Given in the problem is that horses D, E, F and G all have the same probability of winning. Divide the remainder from Step 2 by 4, to find the probability of each of these four horses winning.

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.

Probability Theory
Probability Theory is a fascinating branch of mathematics. It involves the study of random events and the likelihood of various outcomes occurring. In simple terms, it allows us to assign a numerical value to the chance of an event happening.
Given our horse race scenario, each horse's winning probability is an example of a probability assignment.

In this context:
  • Probability of horse A winning is 1/4
  • Probability of horse B winning is 3/10
  • Probability of horse C winning is 1/20
      • Understanding these probabilities helps predict outcomes. The sum of probabilities for all possible outcomes in a probability space must always equal 1. In our case, this ensures that one and only one horse can win the race.
Discrete Mathematics
Discrete Mathematics is an area of mathematics focused on structures that are fundamentally distinct and separate. They are not continuous, like real numbers but involve countable elements.
In the horse race example, the probabilities for different horses form a discrete probability distribution. Each horse can either win or lose, presenting a finite set of possible outcomes.

The key takeaway is that we:
  • Consider discrete, separate outcomes (horse A, B, C, D, E, F, or G winning)
  • Divide a whole into finite, non-overlapping parts to determine individual probabilities
This is a cornerstone of discrete mathematics, making it crucial for tackling problems like our horse race scenario.
Problem Solving
Problem Solving is an essential skill in mathematics, with a focus on identifying solutions to complex questions. Analyzing and methodically breaking down problems into simpler steps enables effective solutions.

Let's outline our problem-solving approach:
  • Identify known probabilities for horses A, B, and C
  • Calculate the remainder of total probability (which is 1) by subtracting known probabilities
  • Allocate the remaining probability equally for the remaining horses D, E, F, and G
Using structured problem-solving ensures that we logically allocate probabilities and verify that all conditions (like total probability equaling 1) are met, making the process both accurate and efficient.

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

Suppose that you roll a pair of honest dice. If you roll a total of \(7,\) you win \(\$ 18\); if you roll a total of 11 , you win \(\$ 54\); if you roll any other total, you lose \(\$ 9 .\) Find the expected payoff for this game.

Imagine a game in which you roll an honest die three times. Find the probability of the event \(E="\) at least one of the rolls of the dice comes up a 6." (Hint: See Example 16.22.)

The ski club at Tasmania State University has 35 members (15 females and 20 males). A committee of three members - a President, a Vice President, and a Treasurer must be chosen. (a) How many different three-member committees can be chosen? (b) How many different three-member committees can be chosen in which the committee members are all females? (c) How many different three-member committees can be chosen in which the committee members are all the same gender? (d) How many different three-member committees can be chosen in which the committee members are not all the same gender?

Bob has 20 different dress shirts in his wardrobe. (a) In how many ways can Bob select seven shirts to pack for a business trip? (b) In how many ways can Bob select 5 of the 7 dress shirts he packed for the business trip \(-\) one for the Monday meeting, one for the Tuesday dinner, one for the Wednesday party, one for the Thursday conference, and one for the Friday date?

There are 500 tickets sold in a raffle. If you have three of these tickets and five prizes are to be given, what is the probability that you will win at least one prize? (Give your answer in symbolic notation.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.