/*! 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 84 In a game of chance, players dra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 84

In a game of chance, players draw one cube out of a bag containing 3 red cubes, 2 white cubes, and 1 blue cube. The player wins \(\$ 5\) if a blue cube is drawn, the player loses \(\$ 2\) if a white cube is drawn. If a red cube is drawn, the player does not win or lose anything. a. Create a table that shows the probability distribution for the amount of money a player will win or lose when playing this game. b. Draw a graph of the probability distribution you created in part a.

Short Answer

Expert verified
For the table, the red cube results in a $0 gain with a 0.5 probability, the white cube results in a $-2 loss with a 0.33 probability, the blue cube results in a $5 win with a 0.17 probability. In the graph, this is visualized with bars of varying heights according to the calculated probabilities for each monetary outcome.

Step by step solution

01

Calculating the Probabilities

First, calculate the probability of drawing each color cube. The total number of cubes is 6 (3 red + 2 white + 1 blue). The probability of drawing a red cube is \(\frac{3}{6} = 0.5\), a white cube is \(\frac{2}{6} =\approx 0.33\), and a blue cube is \(\frac{1}{6} = \approx 0.17\).
02

Creating the Table

Now, create a table to show the monetary outcome and corresponding probability. The table will have three rows (one for each color cube) and two columns (one for the money outcome and one for the corresponding probability). The red cube results in a $0 gain with a probability of 0.5, the white cube results in a $-2 loss with a probability of approx. 0.33, and the blue cube results in a $5 win with a probability of approx. 0.17.
03

Creating the Graph

Finally, graph the probability distribution using a bar chart for visual clarity. The x-axis will represent the monetary gain or loss, and the y-axis will represent the probability. Draw a bar for each cube color at the appropriate height corresponding to the calculated probability. The red cube (0 gain/loss) should be the highest bar (probability 0.5), followed by the white cube (-$2), and the blue cube (+$5).

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 Calculation
When playing games of chance like drawing a cube from a bag, it's important to calculate the probabilities to understand the likelihood of different outcomes. In our example, we have a total of 6 cubes: 3 red, 2 white, and 1 blue. Each draw is equally likely, making the total probability distribution add up to 1. For each color cube:
  • The probability of drawing a red cube is calculated as the number of red cubes (3) divided by the total number of cubes (6). This gives us a probability of: \( P( ext{red}) = \frac{3}{6} = 0.5 \).
  • For the white cubes, the probability is: \( P( ext{white}) = \frac{2}{6} \approx 0.33 \).
  • Finally, for the blue cube, the probability is: \( P( ext{blue}) = \frac{1}{6} \approx 0.17 \).
Calculating these probabilities helps predict the expected outcomes for each draw, such as wins or losses.
Graphing Probability Distributions
Visualizing probability distributions can be immensely helpful to understand how likely each outcome is. Graphs allow us to see the distribution at a glance, making it easier to analyze.
To graph the distribution from our cube game, a bar chart is an effective choice. The x-axis will represent the potential monetary outcomes: $0, $-2, and $5. Meanwhile, the y-axis will show the probability of each outcome.
  • The red cube, which corresponds to no gain or loss (a $0 outcome), will have a bar reaching up to 0.5 on the y-axis, highlighting it as the most probable event.
  • The white cube, resulting in a $-2 loss, will have a bar reaching about 0.33.
  • The blue cube, offering a $5 win, will have the shortest bar, reaching up to 0.17.
Graphing in this manner gives a clear visual representation of which outcomes are more likely, assisting in decision-making and deeper analysis of the game.
Expected Value
Understanding the expected value of a game of chance is crucial to decide if it's worth playing. The expected value gives a theoretical average outcome per play after many repetitions.
To find the expected value, multiply each monetary outcome by its probability and sum these products:
  • For the red cube: \(0 gain times the probability 0.5, yields \( 0 imes 0.5 = 0 \).
  • For the white cube: \)-2 loss times the probability 0.33, yields \( -2 imes 0.33 = -0.66 \).
  • For the blue cube: $5 win times the probability 0.17, yields \( 5 imes 0.17 = 0.85 \).
Adding these results provides the expected value of the game: \[ 0 - 0.66 + 0.85 = 0.19 \].This positive expected value indicates that, on average, a player might gain 19 cents per game over the long run. This calculation helps in understanding the risk and potential reward involved in consistently playing the game.

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

According to a 2017 Gallup poll, \(17 \%\) of Americans report they rarely feel stressed. Suppose 80 Americans are randomly sampled. Find the probability of the following: a. Wxactly 15 rarely feel stressed b. More than 20 rarely feel stressed c. At most 10 rarely feel stressed

he distribution of red blood cell counts is different for men and women. For both, the distribution is approximately Normal. For men, the middle \(95 \%\) range from \(4.5\) to \(5.7\) million cells per microliter and for women, the middle \(95 \%\) have red blood cells counts between \(3.9\) and \(5.0\) million cells per microliter. a. What is the mean and the standard deviation of red blood cell counts for men? Explain your reasoning. b. What is the mean and the standard deviation of red blood cell counts for women? Explain your reasoning. c. Which gender shows more variation in red blood cell counts? Support your answer with appropriate statistics.

Voice-controlled video assistants are being incorporated into a wide variety of consumer products, including smartphones, tablets, and stand-alone devices such as the Amazon Echo or Google Home. A Pew Research poll found that \(46 \%\) of Americans reported using a voice-controlled digital assistant. Suppose a group of 50 Americans is randomly selected. a. Find the probability that more than half of the sample uses a voice- controlled digital assistant. b. Find the probability that at most 20 use a voice-controlled digital assistant. c. In a group of 50 Americans, how many would we expect use one of these devices? d. Find the standard deviation for this binomial distribution. Using your answers to parts \(\mathrm{c}\) and \(\mathrm{d}\), would it be surprising to find that fewer than 10 used one of these devices? Why or why not?

According to the American Veterinary Medical Association, \(30 \%\) of Americans own a cat. a. Find the probability that exactly 2 out of 8 randomly selected Americans own a cat. b. In a random sample of 8 Americans, find the probability that more than 3 own a cat.

Determine whether each of the following variables would best be modeled as continuous or discrete: a. Number of girls in a family b. Height of a tree c. Commute time d. Concert attendance
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.