Imagine you’ve spent months building a basketball-playing robot called the Dunk-O-Matic. You’re thrilled to showcase it at the prestigious Sportecha Conference. However, you stumble upon an advertisement claiming that the Dunk-O-Matic can adjust its skill level to ensure a fair game against any human opponent. This wasn’t part of your original design, which was simply to have the robot take turns shooting baskets with a human player. It seems the CEO might have exaggerated the robot’s capabilities, potentially setting you up for an awkward situation.
Luckily, you included a feature that allows you to tweak the robot’s probability of making a basket on each attempt. You quickly gather information and find that your team has compiled data on all potential participants, including their probabilities of making baskets. In each match, the human player shoots first, followed by the robot, and this continues until one of them successfully makes a basket.
Your task is to adjust the Dunk-O-Matic’s probability so that the human player has a 50% chance of winning each match. At first glance, you might think that the robot’s probability should match the human’s. However, this overlooks the advantage of shooting first. For instance, if both players have a 100% success rate, the first player will always win. This calls for a more detailed analysis.
One effective approach is to use geometric series to calculate the human’s chances of winning. A geometric series is an infinite sum where each term is a multiple of the previous term by a common ratio. Two important facts about geometric series are: if the common ratio’s absolute value is less than 1, the series has a finite sum; and if the first term is ‘a’, the total sum is given by a divided by (1 – r).
In this scenario, the human player has a probability ‘p’ of making a basket. Since they shoot first, they have a probability ‘p’ of winning on their first attempt. The probability of winning on the second attempt occurs only if both players miss their first shots, which has a probability of (1 – p) times (1 – q). The total probability of the human winning can be expressed as a sum of a geometric series.
We want this sum to equal 1/2. By solving for ‘q’, we find that it should equal p divided by (1 – p). If ‘p’ is greater than 50%, then ‘q’ would need to be greater than 1, which is impossible. In such cases, a fair game cannot be achieved because the human has a better than 50% chance of winning immediately.
The robot’s total probability can also be expressed as a geometric series. To win, the robot needs a series of misses followed by a successful shot. By setting ‘q’ to p/(1 – p), we ensure that the series for both the human and the robot have the same sum, meaning they have equal chances of winning.
The demonstration goes smoothly, and while you were worried about potential embarrassment, you also wanted to be honest with the public. Taking the stage, you address your company’s misleading claims and explain your quick solution. Fortunately, the negative press is directed at your employers, and the presentation volunteers represent a more employee-friendly robotics company. After some legal proceedings, you find yourself in a healthier work environment with a regular spot on a pickup basketball team.
Work in pairs to calculate the probability of winning for both the human player and the Dunk-O-Matic using different values of ‘p’. Use the geometric series formula to determine the robot’s probability ‘q’ that ensures a fair game. Present your findings to the class.
Create a simple computer simulation or use a spreadsheet to model a series of basketball games between a human player and the Dunk-O-Matic. Adjust the robot’s probability and observe how it affects the outcome. Share your simulation results and discuss any patterns you notice.
Engage in a debate about the ethical implications of the CEO’s exaggerated claims about the Dunk-O-Matic. Discuss how transparency and honesty in advertising can impact consumer trust and company reputation. Reflect on how you would handle a similar situation.
Explore the concept of geometric series in more depth. Work through examples of geometric series in different contexts, such as finance or physics. Discuss how understanding geometric series can be applied to solve real-world problems.
Role-play the presentation scenario described in the article. One student acts as the engineer explaining the solution, while others play the roles of the audience, including skeptical journalists and supportive colleagues. Practice delivering clear and concise explanations under pressure.
You’ve spent months creating a basketball-playing robot, the Dunk-O-Matic, and you’re excited to demonstrate it at the prestigious Sportecha Conference. However, you come across an advertisement stating, “See the Dunk-O-Matic face human players and automatically adjust its skill to create a fair game for every opponent!” This was not part of your original design; you intended for the robot to shoot baskets with varying success while taking turns with a human opponent. It seems the CEO may have overpromised, potentially setting you up for public embarrassment.
Fortunately, you included a feature that allows you to adjust the robot’s probability of success on each attempt. You quickly gather information and discover that your team has a dossier on all potential demo participants, including their probabilities of making baskets. In each match, the human shoots first, followed by the robot, and this continues until someone makes the first successful basket.
You can remotely adjust the Dunk-O-Matic’s probability for each opponent. The challenge is to determine what that probability should be so that the human has a 50% chance of winning each match.
Initially, one might think that the robot’s probability should equal the human’s, but this overlooks the advantage of going first. For example, if both players have a 100% success rate, the first player will always win. Therefore, a deeper analysis is necessary.
One approach involves using geometric series to calculate the chances of the human winning. A geometric series is an infinite sum where each term is the previous term multiplied by a common ratio. Two key facts about geometric series are: if the common ratio has an absolute value less than 1, the series has a finite total; and if the first term in the series is ‘a’, the total is given by a divided by (1 – r).
To apply this to our situation, the human has a probability ‘p’ of making a basket. Since they shoot first, they have a probability ‘p’ of winning on their first attempt. The probability of winning on the second attempt occurs only if both players miss their first shots, which has a probability of (1 – p) times (1 – q). The total probability of the human winning can be expressed as a sum of a geometric series.
We want this sum to equal 1/2. By solving for ‘q’, we find that it should equal p divided by (1 – p). If ‘p’ is greater than 50%, then ‘q’ would need to be greater than 1, which is not possible. In that case, a fair game cannot be achieved because the human has a better than 50% chance of winning immediately.
The robot’s total probability can also be expressed as a geometric series. To win, the robot needs a series of misses followed by a successful shot. If we set ‘q’ to p/(1 – p), we find that the series for both the human and the robot have the same sum, meaning they have equal chances of winning.
The demonstration goes smoothly, and while you were concerned about potential embarrassment, you also wanted to maintain honesty with the public. Taking the stage, you address your company’s misleading claims and your quick solution. Fortunately, the negative press is directed at your employers, and the presentation volunteers represent a more employee-friendly robotics company. After some legal proceedings, you find yourself in a healthier work environment with a regular spot on a pickup basketball team.
Basketball – A sport that can be modeled mathematically to analyze player performance and game strategies. – In robotics, algorithms can be developed to simulate basketball games and optimize player movements for better scoring opportunities.
Robot – A programmable machine capable of carrying out a series of actions autonomously or semi-autonomously. – The university’s robotics team designed a robot that can solve complex mathematical problems and assist in laboratory experiments.
Probability – A branch of mathematics that deals with the likelihood of occurrence of a given event. – In robotics, probability is used to predict the success rate of a robot navigating through an unknown environment.
Human – A biological entity whose actions and decisions can be modeled and analyzed using mathematical and computational methods. – The interaction between a human and a robot can be optimized by analyzing the human’s behavior patterns and preferences.
Geometric – Relating to the properties and relations of points, lines, surfaces, and solids. – Geometric algorithms are essential in robotics for path planning and object recognition.
Series – A sequence of numbers or functions that follow a particular pattern or rule. – In mathematics, the convergence of a series is crucial for determining the stability of algorithms used in robotics.
Winning – Achieving a desired outcome or goal, often used in the context of optimization problems. – The winning strategy for the robot competition involved a combination of efficient algorithms and precise mechanical design.
Attempts – Efforts made to achieve a particular result, often analyzed statistically to improve performance. – The robot’s attempts to solve the maze were recorded and analyzed to enhance its learning algorithm.
Skills – Abilities or expertise developed through practice and learning, often quantified in robotics for task performance. – The robot demonstrated advanced skills in object manipulation, thanks to its sophisticated control algorithms.
Analysis – The process of examining data or systems to understand their components and relationships. – Mathematical analysis of sensor data is crucial for improving the accuracy and reliability of robotic systems.
