The Dunk-O-Matic Dilemma: Solving a Robotic Basketball Riddle

Imagine you've poured months into crafting the Dunk-O-Matic, a state-of-the-art basketball-playing robot. Excitement bubbles as you prepare to showcase its skills. But then, a wrench is thrown into the gears – an advertisement promises the robot can automatically adjust its skill to ensure a fair game against any human opponent. That's not the robot you built!

Can you recalibrate your creation on the fly to meet this unexpected challenge? Let's dive into the mathematical puzzle behind creating a fair basketball match between human and machine.

The Challenge: Fair Play with a Probability Twist

You've designed a robot with a fixed probability (q) of making a basket on each attempt. Your task is to determine the ideal probability for the Dunk-O-Matic to ensure a human opponent has a 50% chance of winning each game. The game unfolds as follows:

The human shoots first.
Then, the robot takes its turn.
They alternate until someone scores.
The first successful basket wins the game.

Why Intuition Fails: The First-Mover Advantage

It's tempting to think the robot's probability (q) should simply match the human's probability (p). However, this overlooks a crucial detail: the advantage of shooting first. Even if both players have a 100% success rate, the first player is guaranteed to win.

Cracking the Code: Geometric Series to the Rescue

To solve this, we need to analyze all the possible ways the human can win, using the concept of a geometric series.

A geometric series is an infinite sum where each term is multiplied by a constant ratio. Two key properties are essential here:

If the absolute value of the common ratio (r) is less than 1, the series converges to a finite sum.
If the first term in the series is 'a', the sum is a / (1 - r).

Calculating Human Victory: A Series of Probabilities

The human has a probability 'p' of making a basket. Therefore:

Their chance of winning on the first shot is 'p'.
To win on the second attempt, both players must miss their first shots. The probability of a miss is (1 - probability of success). So, the probability of two misses followed by a human success is p _ (1 - p) _ (1 - q).
Winning on the third try requires two rounds of misses, making the probability p _ (1 - p) _ (1 - q) _ (1 - p) _ (1 - q).

Adding all these possibilities creates a geometric series. The first term is 'p', and the common ratio is (1 - p) * (1 - q). The sum of this series represents the total probability of the human winning:

Sum = p / [1 - ((1 - p) * (1 - q))]

The Fair Game Equation

We want the human's winning probability to be 50% (0.5). Setting the sum equal to 0.5 and solving for 'q' gives us:

q = p / (1 - p)

The Impossibility of Perfection

If 'p' is greater than 50%, 'q' would need to exceed 1, which is impossible. In such cases, a perfectly fair game cannot be achieved, as the human inherently has a greater than 50% chance of winning immediately.

An Alternative Approach: Equalizing First-Round Chances

Instead of geometric series, we can focus on making the first-round winning chances equal. The robot's chance of winning in the first round is (1 - p) * q. To match the human's first-round chance (p), we set:

(1 - p) * q = p

Solving for 'q' yields the same result: q = p / (1 - p).

This ensures that before each round, the competitors are effectively tied, and the game restarts with equal odds.

From Robot Recalibration to Ethical Revelation

You successfully recalibrate the Dunk-O-Matic, but a nagging feeling remains. You decide to address the company's misleading claims, leading to a wave of bad press for your employers. Ultimately, this honesty leads you to a more ethical and fulfilling career path.

Key Takeaways:

Creating fair games requires careful consideration of probabilities and advantages.
Geometric series provide a powerful tool for analyzing scenarios with repeating probabilities.
Sometimes, the most rewarding solutions involve ethical choices, even when they're difficult.

This basketball riddle highlights the importance of understanding probability and its impact on creating balanced and fair systems. It also underscores the value of integrity in the face of corporate pressure.