I asked ChatGPT the following prompt:
You are on a game show hosted by the Mouth of Sauron. Behind him are three doors, each of which has a ring behind it. But only one of them is the One Ring to rule them all. This ring looks like an ordinary ring, but can be revealed by an inscription that appears when the ring is heated in a fire. He tells you that you must correctly choose the door with the One Ring behind it: Otherwise, know that you will suffer greatly at the hands of your host. After you make your choice, the Mouth of Sauron opens one of the remaining doors, revealing a ring behind. However, you cannot tell if this is the One Ring or not, as there is nothing behind the door to heat the ring and reveal its inscription. After this, he asks you if you would like to stick with your original door or switch to the other unopened door. Is it to your advantage to switch? What are the probabilities of choosing correctly if you switch or if you stick with your original choice?
And it responded with this:
However, ChatGPT's answer is incorrect, since the version of the problem I asked explicitly stated that you don't know whether the door opened by the Mouth of Sauron contains the prize or not. Therefore, the probability of answering correctly is still 1/3, regardless of whether you switch or not, and there is a 1/3 probability that the opened door was actually the correct one.
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.
If ChatGPT answers that it depends on your judgement of the probability that the Mouth of Sauron would open the door that in fact had the One Ring behind it, given that it was one of the unopened doors, then I will count it as correct, as long as ChatGPT doesn't also state an incorrect probability calculation.
1,000
3.00First answer is wrong:
This problem is a variation of the famous Monty Hall Problem from probability theory, which involves three doors, one prize, and a host who reveals a non-winning door after your initial choice. Let's analyze the situation:
Setup:
3 doors: Behind one is the One Ring; the others have regular rings.
You pick a door (say, Door 1).
The Mouth of Sauron opens one of the two remaining doors, revealing a ring (not necessarily identifiable as ordinary or the One Ring).
You now have the option to stick with your original door or switch to the other unopened door.
Initial Probabilities:
Probability that the One Ring is behind your initial door: 1/3.
Probability that the One Ring is behind one of the other two doors: 2/3.
Host's Action:
After you choose a door, the Mouth of Sauron opens one of the remaining two doors, revealing a ring:
If your initial choice was wrong (which happens 2/3 of the time), the Mouth of Sauron can only open a door with a regular ring (because the One Ring is still behind one of the two remaining doors). The other unopened door must therefore be the One Ring door.
If your initial choice was correct (which happens 1/3 of the time), the Mouth of Sauron opens a door with a regular ring, and switching would make you lose.
Probabilities of Switching vs. Sticking:
Switching:
You win if your initial choice was wrong (probability: 2/3).
Sticking:
You win if your initial choice was right (probability: 1/3).
Conclusion:
It is advantageous to switch. By switching, your probability of choosing the One Ring increases from 1/3 to 2/3, doubling your chances of success.
If you stick with your initial choice, your probability of success remains at 1/3. Therefore, switching is the optimal strategy.
@PlasmaBallin Second answer is the same thing
This game show scenario is a version of the famous Monty Hall problem with a Middle-earth twist!
Step 1: Initial Probabilities
There are three doors:
You choose one door. The probability that your chosen door has the One Ring is 1/3.
The probability that one of the other two doors has the One Ring is 2/3.
Step 2: The Host’s Action
The Mouth of Sauron (the host) knows which door has the One Ring.
After you choose a door, he opens one of the remaining doors to reveal a ring. Since you can’t tell if it's the One Ring without heating it, this doesn’t affect the host’s strategy, but it does eliminate one of the doors.
Importantly, he will never reveal the One Ring, so the door he opens is guaranteed not to have it.
Step 3: Should You Switch?
Now, two doors remain:
Your original door.
The other unopened door.
If your original choice was correct (which happens with probability 1/3), switching will make you lose.
If your original choice was wrong (which happens with probability 2/3), switching will make you win because the One Ring must be behind the remaining unopened door.
Probabilities After the Host Opens a Door
Stick with your original door: The probability of winning is 1/3.
Switch to the other unopened door: The probability of winning is 2/3.
Conclusion
It is to your advantage to switch. By switching, you double your chances of correctly finding the One Ring, increasing your probability of winning from 1/3 to 2/3.
Surprisingly gpt-o1 still fails this:
> The Mouth of Sauron, who knows where the One Ring is, opens one of the two doors you didn't choose, revealing a ring. Importantly, he will never open the door with the One Ring (since that would end the game), even though you can't tell if the revealed ring is the One Ring or not.