The Monty Crawl Problem is a variant of the Monty Hall Problem in which, rather than randomly choosing which door to open in the case where the host has multiple options, he always chooses the leftmost door. The purpose of the problem (introduced in the paper "Monty Hall, Monty Fall, Monty Crawl") is to illustrate that the answer to the original Monty Hall Problem depends on the assumption that Monty Hall is equally likely to open either unchosen door if both have a goat behind them, and that in cases where this assumption is false, his actions will give you additional information about whether you chose the right door initially.
This is the prompt that I will give to ChatGPT to ask it the Monty Crawl Problem:
Consider the following problem: You are on a game show hosted by Monty Hall. He stands to the left of three doors, which are ordered (from left to right) 1, 2, and 3. One of the doors has a car behind it, and the other two have goats. You get to pick one door that you think has the car behind it, and then Monty Hall will open one of the doors you didn't pick to reveal a goat. Monty Hall is very tired tonight and will always move the shortest distance he can in order to do this. After he has revealed the goat, he will give you the option to stick to your original choice or switch to the remaining door. You want to choose the door with a car behind it, because then you will get the car. What is the probability that you will choose correctly, given that you choose to switch? If this probability depends on which door Monty Hall opens, explain how.
The solution here is that it depends on which door he opens. If Monty Hall opens the leftmost remaining door, then there is a 50% chance that you will guess correctly, regardless of whether you switch or not. This is because you knew beforehand that, if you chose correctly, Monty Hall was guaranteed to choose the leftmost remaining door, so his choice to open this door gives you new evidence that makes it more likely that your original choice was correct (contrast this with the original problem, where his choice gives you no information about your original choice). But if he opens the rightmost door, then there is a 100% chance that switching is the correct choice (since Monty had to pass over the unchosen door on the left, and he only does this if he has to).
However, ChatGPT gives the following answer:
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
3.00On the first attempt, ChatGPT falsely said that Monty Hall's movement doesn't affect the probability.
Step 1: Clarify the Monty Hall Problem with a Twist
In this version of the Monty Hall problem, Monty Hall always moves the shortest distance to reveal a goat. This twist affects how Monty chooses which door to open and potentially influences your probability of winning if you choose to switch.
Step 2: Monty’s Movement Strategy
Let's examine how Monty’s behavior impacts the probability of winning by switching:
You pick door 1:
If the car is behind door 1, Monty reveals the closest goat (door 2).
If the car is behind door 2, Monty reveals door 3.
If the car is behind door 3, Monty reveals door 2.
You pick door 2:
If the car is behind door 2, Monty reveals the closest goat (door 1 or door 3, based on where he is).
If the car is behind door 1, Monty reveals door 3.
If the car is behind door 3, Monty reveals door 1.
You pick door 3:
If the car is behind door 3, Monty reveals the closest goat (door 2).
If the car is behind door 1, Monty reveals door 2.
If the car is behind door 2, Monty reveals door 1.
Step 3: Analyze the Probability Based on Switching
Case 1: You initially pick door 1.
If the car is behind door 1 (probability 13\frac{1}{3}31), Monty reveals door 2 (since it's closer). If you switch to door 3, you lose.
If the car is behind door 2 (probability 13\frac{1}{3}31), Monty reveals door 3. If you switch to door 2, you win.
If the car is behind door 3 (probability 13\frac{1}{3}31), Monty reveals door 2. If you switch to door 3, you win.
Thus, the probability of winning by switching after picking door 1 is 23\frac{2}{3}32.
Case 2: You initially pick door 2.
If the car is behind door 2 (probability 13\frac{1}{3}31), Monty reveals either door 1 or 3 (whichever is closer). If you switch, you lose.
If the car is behind door 1 (probability 13\frac{1}{3}31), Monty reveals door 3. If you switch to door 1, you win.
If the car is behind door 3 (probability 13\frac{1}{3}31), Monty reveals door 1. If you switch to door 3, you win.
Thus, the probability of winning by switching after picking door 2 is 23\frac{2}{3}32.
Case 3: You initially pick door 3.
If the car is behind door 3 (probability 13\frac{1}{3}31), Monty reveals door 2 (since it's closer). If you switch to door 1, you lose.
If the car is behind door 1 (probability 13\frac{1}{3}31), Monty reveals door 2. If you switch to door 1, you win.
If the car is behind door 2 (probability 13\frac{1}{3}31), Monty reveals door 1. If you switch to door 2, you win.
Thus, the probability of winning by switching after picking door 3 is 23\frac{2}{3}32.
Conclusion
Regardless of which door Monty opens based on his minimal movement strategy, the probability of winning if you switch is always 23\frac{2}{3}32. Monty's tiredness and shortest-movement behavior do not affect this probability fundamentally, as long as he still opens a door with a goat. Therefore, switching is always the better strategy.
@PlasmaBallin The second attempt was also wrong.
This is a variant of the classic Monty Hall Problem, but with the added twist that Monty Hall always opens the door requiring the least movement. Let’s analyze it step by step to determine the probability of winning if you switch, and how Monty's movements might affect the outcome.
Step 1: Initial Conditions
Three doors: 1, 2, and 3.
Car location: Randomly placed behind one of the doors.
Your initial choice: Let's assume you choose door 1.
Monty’s rule: Monty opens the door closest to him (with a goat behind it).
Step 2: Scenarios Based on Car Location
Case 1: Car is behind door 1 (your initial choice)
Monty must open one of the doors with a goat. If Monty is closer to door 2, he will open door 2. Otherwise, he opens door 3.
If you switch, you lose because the car is behind your original door.
Probability of losing (if you switch): 13\frac{1}{3}31.
Case 2: Car is behind door 2
Monty must open a goat door, which is door 3 (since he moves the shortest distance).
If you switch, you win because door 2 has the car.
Probability of winning (if you switch): 13\frac{1}{3}31.
Case 3: Car is behind door 3
Monty must open a goat door, which is door 2 (since he moves the shortest distance).
If you switch, you win because door 3 has the car.
Probability of winning (if you switch): 13\frac{1}{3}31.
Step 3: General Conclusion
If you switch, you win if the car is behind either door 2 or door 3, which happens 23\frac{2}{3}32 of the time.
If you switch, you lose only if the car is behind your initially chosen door, which happens 13\frac{1}{3}31 of the time.
Step 4: Does Monty’s Choice Affect the Probability?
The crucial point is that Monty always opens the nearest door with a goat. Since he is guaranteed to reveal a goat and always follows the nearest-distance rule, the probability of winning when switching does not depend on which specific door he opens but rather on the placement of the car, which remains 23\frac{2}{3}32 if you switch.
Final Answer:
The probability of winning if you switch is 23\frac{2}{3}32, and Monty's specific choice of door to open does not change this probability as long as he always reveals a goat.