Mining for unobtainium is no easy task. This rare mineral is found in only 1% of the rocks in the mine. Your friend Joe has been working hard on a new unobtainium detector, and it’s finally ready for use. The device is quite reliable, accurately detecting unobtainium when it’s present and giving correct readings 90% of the time otherwise.
On the first day of testing, the detector goes off, and Joe places the rock in his cart. As you both head back to camp to examine the ore, Joe offers to sell you the rock for $200. You know that if the rock contains unobtainium, it would be worth $1000, but any other minerals would be practically worthless. Should you make the trade?
At first, it might seem like a good deal. The detector is mostly accurate, so trusting its reading seems reasonable. However, the situation is more complex than it appears.
Imagine the mine contains exactly 1,000 pieces of ore. With unobtainium making up just 1% of that, there are only 10 rocks with the mineral. All 10 would trigger the detector. But what about the other 990 rocks without unobtainium? About 90% of them, or 891 rocks, won’t trigger the detector. However, 10% of them, or 99 rocks, will trigger the detector despite not containing unobtainium. This is known as a false positive.
This means a total of 109 rocks will have triggered the detector. Joe’s rock could be one of the 10 that contain unobtainium or one of the 99 that don’t, giving it a 10 out of 109 chance—approximately 9%—of containing unobtainium. Paying $200 for a 9% chance of getting $1000 isn’t a great deal.
So why does this result seem surprising, and why did Joe’s rock appear to be a sure thing? The key concept here is the base rate fallacy. While we focus on the detector’s relatively high accuracy, we often overlook how rare unobtainium is. Because the device has a 10% error rate, any positive reading is more likely to be a false positive than a true finding.
This situation illustrates conditional probability. The answer lies not in the overall chance of finding unobtainium or the overall chance of a false positive, but in the chance of finding unobtainium given that the device returned a positive reading. This is known as conditional or posterior probability, determined after narrowing down the possibilities through observation.
Many people struggle with the false positive paradox because we tend to focus on specific information rather than the broader context, especially when making immediate decisions. While it’s often better to be cautious, false positives can lead to real negative consequences. For example, false positives in medical testing can cause unnecessary stress or treatment, and in mass surveillance, they can result in wrongful arrests.
In this case, the one thing you can be sure of is that Joe is trying to take advantage of the situation.
Gather your classmates and simulate the mining scenario using a deck of cards. Assign one suit to represent unobtainium and the others as regular rocks. Shuffle the deck and draw cards to simulate the detector’s readings. Calculate the probability of a card being unobtainium if it triggers the detector. Discuss how this relates to the base rate fallacy.
Work in pairs to solve a set of probability puzzles related to false positives. Use the unobtainium scenario as a starting point and create similar problems with different probabilities and outcomes. Share your puzzles with the class and solve them together, discussing the reasoning behind each solution.
Conduct research on real-world examples of the base rate fallacy and false positives. Present your findings to the class, focusing on how these concepts affect decision-making in fields like medicine, law enforcement, or technology. Highlight the importance of understanding conditional probability in these contexts.
Divide into two groups and debate whether you should buy Joe’s rock. One group argues for the purchase based on the detector’s accuracy, while the other argues against it using the base rate fallacy. After the debate, reflect on how understanding probability can influence decision-making.
Illustrate the unobtainium scenario in a comic strip format. Use humor and creativity to depict the misunderstanding of probabilities and the base rate fallacy. Share your comic strips with the class and discuss how visual storytelling can help explain complex concepts.
Here’s a sanitized version of the provided YouTube transcript:
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Mining unobtainium is challenging. The rare mineral appears in only 1% of the rocks in the mine. However, your friend Joe has been working on an unobtainium detector for months, and it’s finally ready. The device is highly reliable, detecting unobtainium whenever it is present and returning accurate readings 90% of the time otherwise.
On his first day testing it in the field, the device goes off, and Joe places the rock in his cart. As you both head back to camp to examine the ore, Joe offers to sell you the rock for $200. You know that a piece of unobtainium of that size would be worth $1000, but any other minerals would be virtually worthless. Should you make the trade?
Pause here if you want to think it over.
At first glance, it seems like a good deal. Since the detector is mostly accurate, it seems reasonable to trust its reading. Unfortunately, that’s not the case.
Imagine the mine has exactly 1,000 pieces of ore. With unobtainium being 1% of that, there are only 10 rocks containing the mineral. All 10 would trigger the detector. But what about the other 990 rocks without unobtainium? About 90% of them, or 891 rocks, won’t trigger the detector. However, 10% of them, or 99 rocks, will trigger the detector despite not containing unobtainium, which is known as a false positive.
This means that a total of 109 rocks will have triggered the detector. Joe’s rock could be one of those 10 that contain the mineral or one of the 99 that don’t, giving it a 10 out of 109 chance—approximately 9%—of containing unobtainium. Paying $200 for a 9% chance of getting $1000 isn’t a great deal.
So why does this result seem surprising, and why did Joe’s rock appear to be a sure thing? The key concept here is the base rate fallacy. While we focus on the detector’s relatively high accuracy, we often overlook how rare unobtainium is. Because the device has a 10% error rate, any positive reading is more likely to be a false positive than a true finding.
This situation illustrates conditional probability. The answer lies not in the overall chance of finding unobtainium or the overall chance of a false positive, but in the chance of finding unobtainium given that the device returned a positive reading. This is known as conditional or posterior probability, determined after narrowing down the possibilities through observation.
Many people struggle with the false positive paradox because we tend to focus on specific information rather than the broader context, especially when making immediate decisions. While it’s often better to be cautious, false positives can lead to real negative consequences. For example, false positives in medical testing can cause unnecessary stress or treatment, and in mass surveillance, they can result in wrongful arrests.
In this case, the one thing you can be sure of is that Joe is trying to take advantage of the situation.
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This version maintains the core content while removing any informal language or potentially misleading phrases.
Mining – The process of extracting useful information or patterns from large sets of data. – In statistics class, we learned about data mining techniques to analyze survey results and identify trends.
Detector – A tool or algorithm used to identify specific patterns or signals within a set of data. – The teacher explained how a detector can be used in algorithms to find anomalies in a sequence of numbers.
Unobtainium – A hypothetical or fictional material that is extremely rare or impossible to obtain, often used in theoretical discussions. – In our physics project, we joked about needing unobtainium to build a frictionless surface for our experiment.
Trade – The act of exchanging one thing for another, often used in the context of decision-making and resource allocation. – In economics, we discussed how countries trade goods to maximize their comparative advantages.
False – Not true or incorrect, often used in logic and probability to describe an outcome or statement. – The statement that all prime numbers are even is false, as demonstrated by the number 3.
Positive – In mathematics, a value greater than zero; in logic, an affirmative or true statement. – The probability of rolling a positive number on a standard die is 1, since all outcomes are greater than zero.
Probability – The measure of the likelihood that an event will occur, expressed as a number between 0 and 1. – We calculated the probability of drawing an ace from a standard deck of cards as 4 out of 52.
Fallacy – A mistaken belief or error in reasoning, often leading to invalid arguments or conclusions. – The teacher warned us about the gambler’s fallacy, which incorrectly assumes that past events affect future probabilities.
Chance – The occurrence of events in the absence of any obvious intention or cause, often quantified in terms of probability. – The chance of winning the lottery is extremely low, making it an unreliable financial strategy.
Decision – The act of making a choice between different options, often involving analysis and critical thinking. – In our math class, we used decision trees to evaluate different strategies for maximizing profit.
