Question

PROBABILITY. Areas of regions: Entire Target=84in^2, Red area=28in^2, Green area=7in^2, Yellow area=9.42in^2, Blue area=39.58in^2. If you hit the yellow target, you win $2. Hit green target, you win $2. Hit red target, you win $4. Hit anywhere else, you lose $3. Assume it costs $0 to play. What’s the probability and payoff $ amount for red, yellow, green, and blue? What’s the expected value for playing this game? Round result to at least 3 decimal spaces. Is the game fair? Why or why not?

A. Probability: Red = 0.333, Yellow = 0.112, Green = 0.083, Blue = 0.472; Payoff: Red = $4, Yellow = $2, Green = $2, Blue = -$3; Expected Value = -$0.303; The game is not fair because the expected value is negative.
B. Probability: Red = 0.333, Yellow = 0.112, Green = 0.083, Blue = 0.472; Payoff: Red = $4, Yellow = $2, Green = $2, Blue = -$3; Expected Value = $0.303; The game is fair because the expected value is positive.
C. Probability: Red = 0.333, Yellow = 0.112, Green = 0.083, Blue = 0.472; Payoff: Red = $4, Yellow = $2, Green = $2, Blue = -$3; Expected Value = $0; The game is fair because the expected value is zero.
D. Probability: Red = 0.333, Yellow = 0.112, Green = 0.083, Blue = 0.472; Payoff: Red = -$4, Yellow = -$2, Green = -$2, Blue = $3; Expected Value = -$0.303; The game is not fair because the expected value is negative.

Answer 1

To find the probability and payoff amounts for each color, divide the area of each region by the area of the entire target. The expected value is -$0.303, indicating an expected average loss. The game is not fair.

Step-by-step explanation:

To find the probability and payoff amount for red, yellow, green, and blue, we need to divide the area of each region by the area of the entire target. The probabilities are: Red = 28in^2 / 84in^2 = 0.333, Yellow = 9.42in^2 / 84in^2 = 0.112, Green = 7in^2 / 84in^2 = 0.083, and Blue = 39.58in^2 / 84in^2 = 0.472.

The payoffs are: Red = $4, Yellow = $2, Green = $2, and Blue = -$3 (losing $3).

To calculate the expected value, we multiply each probability by its corresponding payoff, and sum all the results. The expected value is -$0.303, rounded to three decimal places.

The game is not fair because the expected value is negative, indicating an expected average loss. Therefore, the correct option is A.