Write a game-type problem about odds and payoffs so that the odds in favor of an event are \( 2: 3 \) and the game is a fair game. Select the correct choice below and fill in the answer box(es) to complete your choice. A. A player pays \( \$ 4.00 \) to spin a spinner with five sectors of equal area. If the spinner lands on one of regions, the player receives \( \$ \square \) B. A player pays \( \$ 4.00 \) to draw a card from a normal 52 -card deck of playing cards. If the suit of the card is a club, the player receives \( \$ \square \) C. A player pays \( \$ 4.00 \) to roll a six-sided die. If the roll is greater than \( \square \), the player receives \( \$ \)

Below is one complete solution written step by step. --- ### Step 1. Determine the Probability of the Event We are told the **odds in favor** of an event are \(2:3\). By definition, if the odds in favor are \(a:b\), then the probability of the event occurring is \[ \frac{a}{a+b}. \] For our event, \[ p=\frac{2}{2+3}=\frac{2}{5}. \] --- ### Step 2. Write a Fair Game Equation A fair game means that the expected winnings are equal to the cost of playing. In our case, the player pays \(\$4.00\). Let \(X\) be the amount the player receives if they win. Because the probability of winning is \(\frac{2}{5}\), we set up the fairness condition as \[ \text{Probability of win} \times X = \$4.00. \] Thus, \[ \frac{2}{5} \times X = 4. \] --- ### Step 3. Solve for \(X\) To solve for \(X\), multiply both sides of the equation by \(\frac{5}{2}\): \[ X = 4 \times \frac{5}{2}= \frac{20}{2}= 10. \] So, the player should receive \(\$10.00\) if the event happens. --- ### Step 4. Match to the Correct Game Type Now let’s consider the given choices: **Choice A.** A player pays \(\$4.00\) to spin a spinner with five sectors of equal area. If the spinner lands on one of regions, the player receives \(\$ \square\). - In this spinner game, if there are five equally sized sectors and the odds in favor are \(2:3\), that implies exactly \(2\) of the \(5\) sectors are "winning" sectors. Thus, the probability of winning is \[ \frac{2}{5}, \] which matches our probability. - We computed the winning amount to be \(\$10.00\). **Choice B** involves drawing a club from a 52‐card deck. However, the probability to draw a club is \[ \frac{13}{52}=\frac{1}{4}, \] which does not match \(\frac{2}{5}\). **Choice C** involves rolling a six-sided die. No matter which number you pick to compare with, the probabilities available (such as \(\frac{2}{6}\) or \(\frac{3}{6}\)) cannot produce exactly \(\frac{2}{5}\). Thus, the correct choice is **Choice A**. --- ### Final Answer **A. A player pays \(\$4.00\) to spin a spinner with five sectors of equal area. If the spinner lands on one of the \(2\) winning regions, the player receives \(\$10.00\).** ---