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Imagine the excitement of unveiling your masterpiece, the Dunk-O-Matic, at the esteemed Sportecha Conference, only to realize that the robot you designed to play basketball has been misrepresented. Instead of a straightforward showdown between man and machine, the robot was expected to adjust its skills to ensure a fair game for every opponent. But how can you create a fair playing field when the rules of the game have changed?
Let's dive into the mathematics of fairness. Suppose you're faced with a challenge: ensuring the human opponent has a 50% chance of winning each match. What probability should the Dunk-O-Matic aim for on each attempt to make this happen?
Pause here and think: What probability, q, should the robot have to ensure a 50% chance of victory for the human?
At first glance, you might assume q should equal p, the human's probability of making a basket. However, this approach fails to account for the advantage of going first. If both had a 100% success rate, the first player would always win. This means a deeper analysis is required.
Enter geometric series, a mathematical tool that can help us solve this problem. A geometric series is an infinite sum of numbers where each term is the previous one multiplied by a common ratio. If the common ratio's absolute value is less than 1, the series converges to a finite total.
The human's probability of winning can be expressed as a geometric series. They have a probability p of winning on the first try. If they miss, and the robot also misses, the probability of the human winning on the second try is p multiplied by (1-p) times (1-q). This pattern continues, creating a series that sums to a total probability of the human winning.
To find q, we need to ensure the sum of this series equals 1/2. Through some algebraic manipulation, we discover that q should be equal to p divided by (1-p). This ensures that the robot's chances of winning are balanced with the human's, creating a fair game.
But what if p is greater than 50%? In this case, q would need to be greater than 1, which is impossible. This means a fair game is unattainable because the human already has a better-than-50% chance of winning immediately.
During the demonstration, you explain the company's false promises and your impromptu solution. The volunteers who participated in the presentation end up working for a more employee-friendly robotics company, and after some legal battles, you find yourself in a better workplace and on a basketball team.
In the end, the Dunk-O-Matic's dilemma highlights the importance of understanding probabilities and finding creative solutions to ensure fairness in unexpected situations. By delving into the mathematics of geometric series, you managed to create a fair game, turning a potential embarrassment into an opportunity for growth and learning.
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