Question

A carnival game has a very large board marked out on a grid of squares so that each square has sides 8.8 cm. A one dollar coin with diameter 2.7 cm is tossed from a short distance onto the grid. If the dollar coin lies entirely in any of the squares, you get it back, plus you win a dollar! If any part of the coin lands on any grid line, you will lose your coin, and thus, lose a dollar.

Note: You can assume that the board is very large so that any dollar coin will land somewhere on the board, and that the lines dividing the squares have an ignorable thickness. You can also assume that the dollar coin is equally likely to land anywhere on the board.

a. Show that this game is not considered to be "fair".

Answer 1

To determine if the carnival game is fair, we calculate that the probability of winning is approximately 0.48, making the probability of losing 0.52. Since the chance of losing is greater than winning, the expected average outcome is negative, indicating that the game is not fair.

Step-by-step explanation:

To determine if the carnival game is fair, we must calculate the probability of winning versus losing. A fair game has an expected value of 0, meaning that there is no advantage or disadvantage to playing the game over the long term.

First, we calculate the area where the coin can land and win. Since the coin has a diameter of 2.7 cm, the radius is 1.35 cm. The area of each square on the board is 8.8 cm x 8.8 cm. However, to land entirely within a square, the center of the coin must land at least 1.35 cm away from any of the borders. This reduces the 'safe' landing area of each square to (8.8 cm - 2 x 1.35 cm) ^ 2 = 6.1 cm x 6.1 cm.

The adjusted area for landing is then 6.1 ^ 2 = 37.21 cm^2, and the original square area is 8.8 ^ 2 = 77.44 cm^2. The probability of winning (P(win)) is the safe area/total area = 37.21 / 77.44 = 0.48, approximately.

The probability of losing is therefore P(lose) = 1 - P(win) = 1 - 0.48 = 0.52. Since the probability of losing is greater than the probability of winning, the expected value is negative. Every time you play the game, on average, you will lose more than you will win, making the game unfair.