A Mathematical Analysis of a Carnival Coin Game
In the realm of carnival games, one intriguing challenge involves a coin and a uniquely designed board. Let's delve into the mechanics and calculations behind a popular carnival game where your objective is to flip a coin and land it in a square without touching its borders.
The Game
The game begins with a coin having a diameter of 13 mm. There is a board with many squares, each measuring 20 mm by 20 mm. Your goal is to flip the coin onto the board and have its center land at least 6.5 mm (half the diameter of the coin) away from all four borders of the square. This ensures that the coin stays entirely within the square.
Breaking Down the Problem
To determine the chances of winning this game, we need to analyze the area where the coin can land without touching the borders. Let's break down the solution step-by-step.
Step 1: Calculate the Radius of the Coin
The given diameter of the coin is 13 mm. Therefore, the radius (r) is:
r 13 mm / 2 6.5 mm
Step 2: Determine the Dimensions of the Square
Each square on the board has dimensions of 20 mm by 20 mm.
Step 3: Calculate the Effective Landing Area for the Coin
For the coin to land within the square without touching the borders, the center of the coin must be at least 6.5 mm away from each border of the square. Thus, we must reduce the dimensions of the square by twice the radius of the coin, once for each side, in consideration of both the top and bottom, as well as the left and right borders.
The adjusted width of the area where the center of the coin can land is:
20 mm - 2 x 6.5 mm 20 mm - 13 mm 7 mm
Therefore, the effective landing area is a square of dimensions 7 mm by 7 mm.
Step 4: Calculate the Area of the Effective Landing Zone
The area (A_{landing}) of the effective landing zone is:
A_{landing} 7 mm x 7 mm 49 mm2
Step 5: Calculate the Area of the Square
The area (A_{square}) of the original square is:
A_{square} 20 mm x 20 mm 400 mm2
Step 6: Calculate the Probability of Winning
The probability (P) of the coin landing in the square without touching the borders is the ratio of the effective landing area to the total area of the square:
P A_{landing} / A_{square} 49 mm2 / 400 mm2 0.1225
Step 7: Convert to Percentage
To express this as a percentage:
P x 100 0.1225 x 100 12.25%
Conclusion
The chances of winning the carnival game by landing the coin in a square without touching the borders is approximately 12.25%. This calculation assumes ideal conditions where the coin lands perfectly and does not roll off the square or land on the edge.
Additional Considerations
To accurately calculate the probability of winning this game, additional factors need to be considered, including the thickness of the coin to determine the likelihood of landing on the edge, the slope of the table to account for rolling, and the size of the board and how the coin is thrown to ensure it lands on the board at all. These factors can affect the actual probability of winning, potentially altering the outcome from the theoretical 12.25% to a more nuanced and practical probability.