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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
The probability distribution table is as follows: \$5 with a probability of 0.17, \$0 with a probability of 0.5, -\$2 with a probability of 0.33. The graph will also show these three points plotted on a coordinate plane.

Step by step solution

01

Identify the possible outcomes and their probabilities

The possible outcomes are drawing a red, white, or blue cube. The probability of each outcome is the number of that color cubes divided by the total number of cubes. Hence, the probability of drawing a red cube is \(\frac{3}{6} = 0.5\), a white cube is \(\frac{2}{6} = 0.33\), and a blue cube \(\frac{1}{6} = 0.17\).
02

Associate each outcome with its monetary value

Drawing a red cube results in no gain or loss (-$0), a white cube results in a loss (-$2), and a blue cube results in a gain ($5).
03

Create a table of probability distribution

For a probability distribution, the table should have the outcome (X) as one column and its probability (P(X)) as the other. The table will be like: \newline X\tP(X) \$5\t0.17 \$0\t0.5 -\$2\t0.33
04

Draw a graph of the probability distribution

Based on the table in Step 3, plot each value of money earned or lost on the x-axis, and the probability of earning or losing that amount of money on the y-axis. On the same graph, plot all the three points: (\$5, 0.17), (\$0, 0.5), (-\$2, 0.33). Then draw a straight line joining these points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Game of Chance
When discussing a game of chance, we refer to any game in which the outcome is strongly influenced by random events. The classic examples of such games include dice rolling, lottery, roulette, and even drawing cubes from a bag as in the exercise provided.

In these games, each possible outcome has a certain probability associated with it, which is calculated based on the total number of possible outcomes. In our cube-drawing game, for instance, the probability of drawing a cube of any particular color is determined by the number of cubes of that color divided by the total number of cubes in the bag.

The fairness or expected value of a game of chance can be evaluated by considering the probabilities of various outcomes along with their respective rewards or consequences. In educational exercises such as this, the main aim is to apply probability theory to real-life scenarios, like betting games, to understand the risks and rewards involved in such activities.
Probability Theory
Probability theory is a branch of mathematics that deals with the likelihood of different outcomes in random processes. The basic principles involve determining the probability of specific events occurring, often represented as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.

The exercise we see here involves a finite probability space with discrete outcomes. By learning to calculate the probability of drawing each color cube, students apply fundamental concepts such as the probability of an event equaling the number of favorable outcomes divided by the total number of possible outcomes. This particular problem also highlights how probability is used in real life to predict financial outcomes based on the likelihood of different events.
Statistical Graph
Statistical graphs are visual representations of data that help us to see patterns and understand the distribution of a dataset more clearly. In our cube-drawing exercise, the graph is a simple way to visualize the probability distribution of the monetary outcome for a player in this game of chance.

The graph plots the monetary outcome (on the x-axis) against the probability (on the y-axis). Each possible monetary outcome, such as winning \(5 or losing \)2, is paired with the probability of that outcome occurring. A correctly drawn graph enables students to quickly comprehend the likelihood of each outcome. This visualization is an important tool in both teaching and learning probabilities because it can clarify concepts that are sometimes counterintuitive when presented in a purely numerical format.

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Most popular questions from this chapter

In a standard Normal distribution, if the area to the left of a z-score is about \(0.6666\), what is the approximate z-score? First locate, inside the table, the number closest to \(0.6666 .\) Then find the z-score by adding \(0.4\) and \(0.03\); refer to the table. Draw a sketch of the Normal curve, showing the area and the \(z\) -score.

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According to Anthropometric Survey data, the distribution of arm spans for females is approximately Normal with a mean of \(65.4\) inches and a standard deviation of \(3.2\) inches. a. What percentage of women have arm spans less than 61 inches? b. Olympic swimmer, Dana Vollmer, won a bronze medal in the 100-meter butterfly stroke at the 2016 Olympics. An estimate of her arm span is 73 inches. What percentage of females have an arm span at least as long as Vollmer's?

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