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Chapter 11: Problem 18

Involve computing expected values in games of chance. The spinner on a wheel of fortune can land with an equal chance on any one of ten regions. Three regions are red, four are blue, two are yellow, and one is green. A player wins \(\$ 4\) if the spinner stops on red and \(\$ 2\) if it stops on green. The player loses \(\$ 2\) if it stops on blue and \(\$ 3\) if it stops on yellow. What is the expected value? What does this mean if the

Short Answer

Expert verified
The expected value is \$0 (since \$1.2+\$0.2-\$0.8-\$0.6=\$0). That means, on average, the player neither wins nor loses money each time they play this game.

Step by step solution

01

Determination of Probabilities

Firstly, determine the probabilities of landing on each color. The probability (\(P\)) of an event is calculated as the number of ways an event can happen (\(n\)) over the total number of outcomes (\(N\)), i.e, \(P= n/N\). There are 10 regions on the spinner, so \(N=10\). The probabilities are as: \(P_{red}= 3/10\), \(P_{blue}= 4/10\), \(P_{yellow}= 2/10\), and \(P_{green}= 1/10\).
02

Calculation of Expected Value

Expected Value (E) is calculated as the sum of the product of each outcome and its respective probability. In symbols, \(E= \sum (outcome \times probability)\). For each color, multiply its associated value by the probability calculated in step 1. Red: \(\$4 \times 3/10=$1.2\), Green: \(\$2 \times 1/10=$0.2\), Blue: \(\$-2 \times 4/10=-$0.8\), Yellow: \(\$-3 \times 2/10=-$0.6\). Adding these gives the expected value.
03

Interpretation of Result

The result is the expected gain or loss from each time the game is played. If the expected value is positive, it means the player could expect to win every time they play the game. If it's negative, the player can expect to lose every time the game is played.

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Key Concepts

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

Probability Calculation
Calculating probability is the cornerstone of understanding any game of chance. In essence, probability helps us determine how likely an event is to happen. To calculate probability, you take the number of specific outcomes you're interested in, and divide it by the total number of possible outcomes. For the wheel of fortune game mentioned, the wheel is divided into 10 equal regions.
This means the total number of outcomes, \(N\), is 10.
  • If the spinner has 3 red sections, the probability of it landing on a red is \(\frac{3}{10}\).
  • For blue sections, because there are 4, the probability is \(\frac{4}{10}\).
  • Similarly, there are 2 yellow sections, leading to a probability of \(\frac{2}{10}\).
  • Finally, there's only 1 green section, so the probability for green is \(\frac{1}{10}\).
Understanding these probability ratios is key to advancing further to the expected value calculation.
Wheel of Fortune Probabilities
A wheel of fortune not only provides excitement but is an excellent case study of applied probabilities. Each segment or color on the wheel has an equal chance when spun. This makes it a great example to explore outcomes in a set, known as a sample space. Every spin of the wheel is independent, meaning past spins don't affect future spins. This is true for fair games.
The uniformity of the probabilities, like 3 red sections out of a total of 10 sections, makes it an interesting exercise for calculating likelihoods. By understanding these basics, players and learners can start to discern how games are constructed and the varying possibilities that come with them.
Games of Chance Outcomes
In games of chance, each possible outcome comes with its own consequence. These outcomes are weighted by their probabilities. This is where the calculation of gains and losses comes in. In the wheel of fortune exercise, each color corresponds to a specific monetary result:
  • Landing on red gives you \(\\(4\)
  • Stopping on green results in \(\\)2\)
  • If you hit a blue, you lose \(\\(2\).
  • Rolling yellow costs you \(\\)3\).
These consequences, alongside their probabilities, allow for the calculation of expected gain or loss. The true intrigue lies in using this information strategically, whether for fun or profit.
Expected Gain or Loss
The expected value gives players a long-term average of what they can expect to gain or lose per spin. It's calculated by multiplying each outcome by its probability and summing these products. In our wheel of fortune example:
  • Red: \((4 \, \times \, \frac{3}{10} = \\(1.2)\)
  • Green: \((2 \, \times \, \frac{1}{10} = \\)0.2)\)
  • Blue: \((-2 \, \times \, \frac{4}{10} = -\\(0.8)\)
  • Yellow: \((-3 \, \times \, \frac{2}{10} = -\\)0.6)\)
Adding these gives an overall expected value, which in this case sums to \(0\$.\) This means that over numerous spins, you neither gain nor lose money on average per spin. Understanding this concept allows players to predict the fairness and sustainability of a game.

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Most popular questions from this chapter

In Exercises 31-36, consider a political discussion group consisting of 5 Democrats, 6 Republicans, and 4 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting two Democrats.

A popular state lottery is the \(5 / 35\) lottery, played in Arizona, Connecticut, Illinois, Iowa, Kentucky, Maine, Massachusetts, New Hampshire, South Dakota, and Vermont. In Arizona's version of the game, prizes are set: First prize is \(\$ 50,000\), second prize is \(\$ 500\), and third prize is \(\$ 5\). To win first prize, you must select all five of the winning numbers, numbered from 1 to 35 . Second prize is awarded to players who select any four of the five winning numbers, and third prize is awarded to players who select any three of the winning numbers. The cost to purchase a lottery ticket is \(\$ 1\). Find the expected value of Arizona's "Fantasy Five" game, and describe what this means in terms of buying a lottery ticket over the long run.

Involve computing expected values in games of chance. Another option in a roulette game (see Example 6 on page 760 ) is to bet \(\$ 1\) on red. (There are 18 red compartments, 18 black compartments, and 2 compartments that are neither red nor black.) If the ball lands on red, you get to keep the \(\$ 1\) that you paid to play the game and you are awarded \(\$ 1\). If the ball lands elsewhere, you are awarded nothing and the \(\$ 1\) that you bet is collected. Find the expected value for playing roulette if you bet \(\$ 1\) on red. Describe what this number means.

In a class of 50 students, 29 are Democrats, 11 are business majors, and 5 of the business majors are Democrats. If one student is randomly selected from the class, find the probability of choosing a Democrat or a business major.

In a product liability case, a company can settle out of court for a loss of \(\$ 350,000\), or go to trial, losing \(\$ 700,000\) if found guilty and nothing if found not guilty. Lawyers for the company estimate the probability of a not-guilty verdict to be \(0.8\). a. Find the expected value of the amount the company can lose by taking the case to court. b. Should the company settle out of court?
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