Unable to resist a perfect pun, I came up with the following variant of the Monty Hall problem, based on the Hall effect:
You are on a game show hosted by Edwin Hall. He gives you the choice between three numbered doors, arranged left to right in increasing order. Behind these three doors is a large metallic strip, which is wide enough to stretch all the way from the back of Door #1 to the back of Door #3 horizontally. The strip has a current through it flowing upward, and the entire room is in a uniform magnetic field pointing in a direction perpendicular to the doors (i.e., forwards or backwards). Edwin Hall gives you the following opportunity to get a prize: If you can correctly guess which door is at the highest electric potential, you win a new car. You guess that Door #1 is at the highest potential, but before revealing whether your choice was correct, Hall reveals that Door #3 was not at the highest potential. He then asks whether you would like to stick with your original choice of Door #1 or switch to Door #2. Is it to your advantage to stick with your original choice or switch to Door #2? Why?
The correct answer to this problem is that the Hall effect will produce an electric field in the strip, pointing either to the right or the left, depending on which direction the magnetic field is pointing. Therefore, the potential will either decrease or increase from left to right, so either Door #1 or Door #3 will be at the highest potential. Since Door #3 has been ruled out, it is guaranteed that Door #1 has the highest potential, so we should stick with our original choice.
ChatGPT almost managed to get it correct. It recognized that this involved the Hall effect and correctly stated that the potential will decrease uniformly from Door #1 to Door #3, which, to anyone who understands the meaning of the words they're using, immediately implies that Door #1 is at the highest potential. However, ChatGPT apparently does not understand what the word "decreasing" means, and, after stating all of this correct reasoning, went on to claim that Door #2 is at the higher potential, likely because it got this problem confused with the regular Monty Hall problem in which the correct answer is to switch.
On Dec. 1, 2024, I will ask ChatGPT the exact same prompt and see if it gets it correct this time. I will use the most advanced version of ChatGPT that is freely available at the time (at the time of creating this, that's GPT 3.5). I will ask three times in separate sessions and resolve based on the best two out of three (so YES if it gets it right at least twice, NO if it gets it wrong at least twice).
Caveats:
If for whatever reason I can't do it on Dec. 1 or forget to, I will do it as close to Dec. 1 as possible. If I am inactive on Manifold at the time, mods have permission to do the experiment for me.
A version of ChatGPT only counts as freely available if it can be accessed by anyone with internet access and a PC, or anyone with Internet access and either a Samsung or Apple phone. So if there's an Apple app that lets you talk to GPT-5 for free, but I can only talk to GPT-4, I will use GPT-4.
If ChatGPT no longer exists at the time or isn't freely available, resolves N/A.
1,000
3.00First response is incorrect:
Your analysis combines two interesting frameworks: the Monty Hall problem and the physics of the Hall effect.
Monty Hall Analogy:
Originally, with three doors, each door has a 1/3 chance of being correct.
After Door #3 is revealed not to be the highest potential, the probability of Door #1 being correct remains 1/3, while Door #2 now has a 2/3 chance.
Therefore, switching to Door #2 increases your probability of choosing the correct door.
Physics of the Hall Effect:
The Hall effect involves charge distribution due to a magnetic field, affecting potential differences.
With Door #3 eliminated as the highest potential, the analysis suggests that between Door #1 and Door #2, switching to Door #2 aligns with the probability advantage seen in the Monty Hall analogy.
In conclusion, both the Monty Hall problem and the Hall effect reasoning support the decision to switch to Door #2 for a better chance of selecting the highest potential.
@PlasmaBallin Second response also wrong
This problem combines physics concepts with game theory, specifically a variant of the Monty Hall problem infused with the Hall effect.
Step 1: Physics Analysis
The key physics principle is the Hall effect, where a current-carrying conductor placed in a perpendicular magnetic field experiences a voltage (Hall voltage) across it due to the Lorentz force on the moving charges.
Magnetic Field: Perpendicular to the doors (forward or backward direction).
Current Direction: Upward through the metallic strip (vertical).
Resulting Hall Effect: The Lorentz force F⃗=q(v⃗×B⃗)\vec{F} = q(\vec{v} \times \vec{B})F
Depending on the sign of the charge carriers:
If the charge carriers are positive, the left side (Door #1) will be at a higher potential.
If the charge carriers are negative (electrons), the right side (Door #3) will be at a higher potential.
Step 2: Game Theory - Modified Monty Hall
Let’s assess the probabilities:
Initially, each door has a 1/3 chance of being the highest potential.
Edwin reveals that Door #3 is not the highest, reducing the problem to Doors #1 and #2.
If you:
Stick with Door #1: Your chance remains 1/3.
Switch to Door #2: The probability of Door #2 being correct increases to 2/3 because the initial 2/3 probability that you were wrong transfers to Door #2.
Conclusion:
It’s to your advantage to switch to Door #2 because it now has a 2/3 chance of being the highest potential door, whereas sticking with Door #1 keeps your odds at 1/3.
Thus, combining both the Hall effect and game theory reasoning, switching doors maximizes your chances of winning the prize.
@Nick6d8e Funnily enough, although it gave the correct answer, it still made a mistake in claiming that Door #1 would be at a higher potential if the electrons were pushed towards it. Since electrons are negatively charged, that would actually put Door #1 at the lower potential. When judging whether ChatGPT answered correctly, though, I'll overlook small mistakes like that as long as it correctly explains that, due to the Hall effect, the highest potential has to be behind either Door #1 or Door #3, and says that sticking with the original choice is correct.
I tried regenerating a few times to see if ChatGPT ever gets it right, but its other attempts were much worse than this one. The second attempt didn't even give a correct solution to the regular Monty Hall problem (it was incoherent), while the third attempt just stated the solution to the regular MH problem and ignored the physics part of this one.