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 \( \$ \)

Let’s take a closer look at each option to determine how the payoffs might work out to maintain fairness in the game. A. In this scenario, if the odds in favor of landing on a winning sector are 2:3, there are 5 equal sectors, which implies 2 winning sectors and 3 losing sectors. The player pays \$4.00, and to ensure the game is fair, the payoff must return some amount that equals the potential loss plus the cost of playing. Let's denote the winning payoff as \(x\). The expected value equation would be: \[ \frac{2}{5} x - \frac{3}{5}(4) = 0 \implies 2x - 12 = 0 \implies x = 6. \] So, if the spinner lands on a winning sector, the player would receive \( \$6.00 \). B. With a 52-card deck, the probability of drawing a club (there are 13 clubs) is \( \frac{13}{52} \) or \( \frac{1}{4} \), which translates to odds of \( 1:3 \). To maintain fairness, we conduct a similar expected value calculation. If the bonus payout for a winning card (a club) is represented by \(y\): \[ \frac{1}{4}y - \frac{3}{4}(4) = 0 \implies y - 12 = 0 \implies y = 12. \] Therefore, if a player draws a club, the payout should be \( \$12.00 \). C. For the die roll, the payoff depends on the number rolled. If the player receives a payout for rolling greater than a certain number, say \(z\), the odds need to correspond to the outcomes. For example, if the roll should be greater than 4, only 2 outcomes (5 and 6) are favorable, giving 2 winning and 4 losing results, yielding odds of \( 1:2 \). Following previous calculations, let’s determine the payout \(w\): \[ \frac{1}{3}w - \frac{2}{3}(4) = 0 \implies w - 8 = 0 \implies w = 8. \] If the player rolls greater than 4, they receive \( \$8.00 \). Feel free to use any of these arrangements to craft your game in a way that’s engaging and fair!