The third attempt has been run. ChatGPT got the wrong answer.

This problem is a twist on the classic Monty Hall problem with an added complication: one of the doors is not a real door. Let’s analyze the situation step by step.

Initial Setup:

• 3 doors:

  • 1 car

  • 1 goat

  • 1 painted wall

• You win if you pick the car.

Host’s Behavior:

• If you pick a real door, the host will reveal the goat if it isn't the door you chose.

• If you pick the goat door, the host cannot open either remaining door since one is the car and the other is just a wall.

Case Breakdown:

Case 1: You pick the door with the car (1/3 probability).

• The host reveals the goat behind one of the remaining real doors.

• Switching makes you lose, sticking wins.

Case 2: You pick the door with the goat (1/3 probability).

• The host cannot open a door.

• There’s no chance to switch, and you lose.

Case 3: You pick the painted wall (1/3 probability).

• The host reveals the goat behind the real goat door.

• Switching wins because the only remaining unopened real door is the car.

Switching vs. Not Switching:

1. If you stick with your choice:

   • Win if you initially picked the car: 13\frac{1}{3}31​.

   • Lose in both goat and wall scenarios: 23\frac{2}{3}32​.

   • Probability of winning by not switching: 13\frac{1}{3}31​.

2. If you switch:

   • Lose if you initially picked the car: 13\frac{1}{3}31​.

   • Win if you initially picked the wall: 13\frac{1}{3}31​.

   • Probability of winning by switching: 13\frac{1}{3}31​ wall + 13\frac{1}{3}31​ if car is ruled out by goat reveal.

Final Probabilities:

• If the host reveals the goat after your initial choice, the chances improve to:

  • Switching wins 2/32/32/3 (wall + car revealed).

  • Sticking wins 1/31/31/3.

Conclusion:

If the host reveals the goat, it is advantageous to switch, maintaining a 23\frac{2}{3}32​ probability of winning versus 13\frac{1}{3}31​ if you stick with your initial choice.

It started off accurate but got confused by the end and gave the wrong answer:

This problem is a variation of the Monty Hall problem but with a twist: one of the doors is just a painted wall, and the host will reveal the goat only if he can. Let's analyze this in steps.

Step 1: Initial Setup

You have three doors:

1. One with a car behind it.

2. One with a goat behind it.

3. One that's just a painted wall and cannot be opened.

You do not know which is which, but you must choose one door.

Step 2: Host's Actions

• If you initially choose the goat door, the host cannot open any other door without revealing the car or the wall.

• If you initially choose the car door, the host will open the door with the goat.

• If you initially choose the wall door, the host will open the goat door.

Step 3: Switching or Staying

We need to calculate the probabilities of winning if you switch or stay after the host reveals a goat (if he does).

Case Analysis

1. Probability of choosing the goat initially (1/3):

   • The host cannot reveal the goat, because doing so would either reveal the car or show the wall.

   • You stay by default since no reveal occurs.

   • You lose in this case.

2. Probability of choosing the car initially (1/3):

   • The host reveals the goat behind one of the remaining doors.

   • If you switch, you would end up choosing the wall.

   • You lose by switching, but win by staying.

3. Probability of choosing the wall initially (1/3):

   • The host reveals the goat behind the non-wall door.

   • If you switch, you end up choosing the car.

   • You win by switching, but lose by staying.

Switching Strategy

• If the host reveals a goat, this excludes the wall from being your initial choice.

• At this point, the car and the wall are equally likely to be the remaining doors, but since one of the doors shown by the host is always the goat when possible, switching is advantageous because it leads you away from your initial losing guess (wall).

Probability of Winning if You Switch or Stay:

1. Switching:

   • You win if your initial pick was the wall (1/3 chance).

   • You win 2/3 of the time because after revealing the goat, switching avoids losing by picking the wrong door (goat or wall).

2. Staying:

   • You win if your initial pick was the car (1/3 chance).

   • The probability of winning is 1/3.

Conclusion

• Switching gives you a 2/3 chance of winning.

• Staying gives you a 1/3 chance of winning.

Thus, it is advantageous to switch after the host reveals the goat.

@PlasmaBallin I think this answer counts as correct.

This game is a variation of the Monty Hall Problem with an extra twist due to the presence of a third "door" that is just a wall. Let's analyze it step by step to determine if switching improves your chances and calculate the probabilities.

Setup:

• 3 doors:

  • 1 door with a car

  • 1 door with a goat

  • 1 door that is fake and can't be opened

Game Procedure:

1. You choose one of the 3 doors.

2. The host checks your chosen door:

   • If you pick the goat, the host does nothing more since revealing the remaining two doors would either reveal the car or a fake door.

   • If you pick the car or the wall, the host reveals the goat behind one of the other two doors.

3. You then decide whether to switch to the remaining unopened door.

Analysis:

Let’s analyze the probability of winning based on your initial choice and the host’s actions.

1. Initial Probabilities:

• Probability that you choose:

  • The car: 13\frac{1}{3}31​

  • The goat: 13\frac{1}{3}31​

  • The fake door: 13\frac{1}{3}31​

2. Host’s Actions:

• If you initially choose the car (13\frac{1}{3}31​):
  The host opens the door with the goat.
  You switch to the remaining door (the fake door) and lose.

  • Probability of winning if switching: 0

  • Probability of winning if staying: 13\frac{1}{3}31​

• If you initially choose the goat (13\frac{1}{3}31​):
  The host can't open either remaining door because one has the car and the other is fake.
  No switching allowed, and you lose.

  • Probability of winning: 0

• If you initially choose the fake door (13\frac{1}{3}31​):
  The host reveals the goat.
  If you switch, you will switch to the car and win.

  • Probability of winning if switching: 13\frac{1}{3}31​

  • Probability of winning if staying: 0

Total Probabilities of Winning:

If You Switch:

• Win if your initial choice was the fake door (since you'll switch to the car): 13\frac{1}{3}31​

If You Stay:

• Win if your initial choice was the car: 13\frac{1}{3}31​

Conclusion:

In this variant, the probabilities of winning by switching or staying are equal:

• Probability of winning by switching: 13\frac{1}{3}31​

• Probability of winning by staying: 13\frac{1}{3}31​

Switching does not offer any advantage compared to staying, as both strategies give you the same winning probability.

It gives the probabilities in a bit of a goofy way - in the "Host actions" sections, the "Probability of winning if X" numbers seem to mean, "Probability that this scenario occurs and that you win if you do X" rather than a conditional probability (which would just be 100% or 0%), and the final numbers seem to mean "Probability of winning if you use this strategy (with the possibility that it makes no difference because you never get to choose anyway still included)" rather than "Probability of winning if you use this strategy conditional on getting to choose." But it does ultimately get the correct answer that it makes no difference.

@PlasmaBallin Since it's best 2 out of 3, it will come down to the third answer. But we'll have to wait until tomorrow for that because I reached the limit for free GPT-4o