Problem 20 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

WHEEL OF FORTUNE A wheel of fortune is divided into 20 circular sectors of equal area. The wheel is spun and a payoff is made according to the color of the region on which the indicator lands. One region is gold and pays \(\$ 50\) when hit; five regions are blue and pay \(\$ 25\); and four other regions are red and pay \(\$ 10\). The remaining 10 regions are black and pay nothing. Would you be willing to pay \(\$ 10\) to play this game?

Step-by-Step Solution

Answer

Yes, because the expected payoff (\(\text{\textdollar}10.75\)) is greater than the cost to play (\(\text{\textdollar}10\)).

Step 1: Determine the Probability of Each Outcome

Identify the probability for the indicator to land on each color. There are 20 sectors in total.- Gold: 1 sector, so the probability is \(\frac{1}{20}\).- Blue: 5 sectors, so the probability is \(\frac{5}{20} = \frac{1}{4}\).- Red: 4 sectors, so the probability is \(\frac{4}{20} = \frac{1}{5}\).- Black: 10 sectors, so the probability is \(\frac{10}{20} = \frac{1}{2}\).

Step 2: Calculate the Expected Payoff

Calculate the expected payoff using the formula \(\text{Expected Payoff} = P(\text{Gold}) \times \text{Payoff(Gold)} + P(\text{Blue}) \times \text{Payoff(Blue)} + P(\text{Red}) \times \text{Payoff(Red)} + P(\text{Black}) \times \text{Payoff(Black)}\)Substitute the values:Expected Payoff = \(\frac{1}{20} \times 50 + \frac{5}{20} \times 25 + \frac{4}{20} \times 10 + \frac{10}{20} \times 0\) = 2.5 + 6.25 + 2 + 0 = 10.75

Step 3: Compare the Expected Payoff to the Cost to Play

Compare the expected payoff to the cost of playing the game (\(\text{\textdollar}10\)).The expected payoff is \(\text{\textdollar}10.75\), which is greater than \(\text{\textdollar}10\).

Step 4: Make a Decision

Since the expected payoff (\(\text{\textdollar}10.75\)) is greater than the cost to play (\(\text{\textdollar}10\)), it is beneficial to play the game.

Key Concepts

Probability

Probability is the chance that a certain event will occur. For the wheel of fortune, we first need to find the probabilities of the wheel landing on different colors.
Since there are 20 sectors in total, we calculate the probability for each color by dividing the number of sectors of that color by the total number of sectors:

• Gold: 1 sector, so the probability is \(\frac{1}{20}\).
• Blue: 5 sectors, so the probability is \(\frac{5}{20} = \frac{1}{4}\).
• Red: 4 sectors, so the probability is \(\frac{4}{20} = \frac{1}{5}\).
• Black: 10 sectors, so the probability is \(\frac{10}{20} = \frac{1}{2}\).

These probabilities are essential to understanding the chances of landing on a specific color and the corresponding payout.
Being able to calculate probabilities helps in many areas such as games, investments, and everyday decision-making.

Payoff

The payoff refers to the reward received from landing on a specific color. For each spin of the wheel of fortune, landing on different colors results in different payoffs. The payoffs are:

• Gold: \(\text{\textdollar}50\)
• Blue: \(\text{\textdollar}25\)
• Red: \(\text{\textdollar}10\)
• Black: \(\text{\textdollar}0\)

The payoffs are a key component in calculating the expected value of each spin. They indicate the potential earnings and help in evaluating whether the game is worth playing. Understanding payoffs will enhance your ability to make financial decisions and analyze risk-benefit scenarios.

Expected Payoff Calculation

The expected payoff is the average amount you can expect to win per play in the long run. To calculate the expected payoff, you need to multiply each payoff by the probability of its occurrence and then sum these products:
\[ \text{Expected Payoff} = P(\text{Gold}) \times \text{Payoff(Gold)} + P(\text{Blue}) \times \text{Payoff(Blue)} + P(\text{Red}) \times \text{Payoff(Red)} + P(\text{Black}) \times \text{Payoff(Black)} \]
Using the probabilities and payoffs from the wheel of fortune:
\[ \text{Expected Payoff} = \frac{1}{20} \times 50 + \frac{5}{20} \times 25 + \frac{4}{20} \times 10 + \frac{10}{20} \times 0 = 2.5 + 6.25 + 2 + 0 = 10.75 \]
This means that on average, you can expect to win \(\text{\textdollar}10.75\) per game.
Understanding this concept aids in making informed decisions about whether to engage in activities involving risk, such as gambling or investments.

Decision Making

Decision making involves comparing the expected payoff to the cost of playing the game. In this scenario, the cost to play is \(\text{\textdollar}10\). Since the expected payoff is \(\text{\textdollar}10.75\), which is higher than the cost, it is beneficial to play the game.
This principle can be applied to various situations in life. By weighing the potential benefits against the costs, you can make more informed decisions.
Effective decision making is a valuable skill, and understanding the role of probabilities and expected values is crucial in achieving better outcomes.