91Ó°ÊÓ

Suppose that a certain casino has the "money wheel" game. The money wheel is divided into 50 sections, and the wheel has an equal probability of stopping on each of the 50 sections when it is spun. Twenty-two of the sections on this wheel show a \$1 bill, 14 show a \(\$ 2\) bill, 7 show a \(\$ 5\) bill, 3 show a \(\$ 10\) bill, 2 show a \(\$ 20\) bill, 1 shows a flag, and 1 shows a joker. A gambler may place a bet on any of the seven possible outcomes. If the wheel stops on the outcome that the gambler bet on, he or she wins. The net payofts for these outcomes for \(\$ 1\) bets are as follows. $$ \begin{array}{l|rrrrrrr} \hline \text { Symbol bet on } & \$ 1 & \$ 2 & \$ 5 & \$ 10 & \$ 20 & \text { Flag } & \text { Joker } \\ \hline \text { Payoff (dollars) } & 1 & 2 & 5 & 10 & 20 & 40 & 40 \\ \hline \end{array} $$ a. If the gambler bets on the \(\$ 1\) outcome, what is the expected net payoff? b. Calculate the expected net payoffs for each of the other six outcomes. c. Which bet(s) is (are) best in terms of expected net payoff? Which is (are) worst?

Step by step solution

01

Identify Important Variables

Identify the number of each type of outcome and their corresponding payoffs: \(\$1\) outcome (22 sections, \$1 payoff), \(\$2\) outcome (14 sections, \$2 payoff), \(\$5\) outcome (7 sections, \$5 payoff), \(\$10\) outcome (3 sections, \$10 payoff), \(\$20\) outcome (2 sections, \$20 payoff), flag outcome (1 section, \$40 payoff), joker outcome (1 section, \$40 payoff).

02

Calculate the Probability of Outcomes

Determine the probability of each outcome. As there are 50 sections on the wheel, the probability for each outcome is its count divided by the total number of outcomes (50). For example, the probability of a \(\$1\) outcome is \(22/50\). Calculate the probability for each outcome.

03

Find Expected Payoffs for Each Bet

To find the expected payoff for each bet, multiply the probability of that outcome by its net payoff. For example, to calculate the expected payoff for a \(\$1\) bet: \(Probability(\$1) * Payoff(\$1) = \frac{22}{50} * 1\). Repeat this process for all seven outcomes.

04

Compare Expected Payoffs

Compare the expected payoffs for each bet. The bet with the highest expected payoff is the best in terms of potential gain. The bet with the lowest expected payoff is the worst.

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Key Concepts

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

Expected Net Payoff

In probability, the expected net payoff represents the average amount a gambler can expect to win or lose per bet if they play a game many times. It is calculated by multiplying the probability of each outcome by their respective net payoffs and then summing up these values. This gives a sense of whether an action, like placing a specific bet, is favorable in a game over the long term.

For example, in the money wheel game, if a gambler bets on the \(\$1\) outcome, the formula to calculate the expected net payoff would be: \[ \text{Expected Net Payoff} = \left( \frac{22}{50} \right) \times 1 \] This means the gambler can, on average, expect a net gain or loss from betting on this outcome.

Calculating the expected net payoff for all potential bets lets the gambler make informed decisions about which bets offer the best returns and which to avoid because of higher losses.

Outcomes

Outcomes are the possible results that can occur when playing a game or performing an experiment. In the context of the casino money wheel, outcomes are represented by the sections the wheel could land on: \(\\(1\) bill, \(\\)2\) bill, \(\\(5\) bill, \(\\)10\) bill, \(\\(20\) bill, a flag, and a joker.

Each outcome has a specific frequency, represented by how many sections of the wheel that symbol occupies. For instance, the \(\\)1\) outcome appears 22 times on the wheel, indicating that \(\$1\) bills take up 22 sections. Similarly, other symbols appear based on their designated sections.The crucial aspect of outcomes in probability and gambling involves understanding how often an outcome occurs, which is used to determine probabilities that influence expected payoffs.

Probability Distribution

Probability distribution refers to how probabilities are assigned to different outcomes in an event or experiment. In the case of the money wheel, each outcome has a specific probability calculated by dividing the number of sections it occupies by the total sections on the wheel.

• The probability of landing on a \(\\(1\) bill: \(\frac{22}{50}\)
• The probability of landing on a \(\\)2\) bill: \(\frac{14}{50}\)
• The probability of landing on a \(\\(5\) bill: \(\frac{7}{50}\)
• The probability of landing on a \(\\)10\) bill: \(\frac{3}{50}\)
• The probability of landing on a \(\$20\) bill: \(\frac{2}{50}\)
• The probability of a flag: \(\frac{1}{50}\)
• The probability of a joker: \(\frac{1}{50}\)

This distribution helps assess how likely each outcome is to occur when the wheel spins. Understanding the probability distribution of a game is vital as it provides insight into the likely results and is foundational for calculating expected payoffs.