Bigloop Doors Paradox

Caroline Mae

2 min read

·

05-01-2025

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The Bigloop Doors Paradox, a twist on the classic Monty Hall problem, presents a fascinating exploration of probability and conditional probability. While seemingly simple, its solution often defies intuition, leading to lively debates among mathematicians and puzzle enthusiasts. Let's delve into this intriguing paradox and unravel its surprising answer.

Understanding the Setup

Imagine three doors: Door A, Door B, and Door C. Behind one door is a brand new car; behind the other two are goats. You choose a door, say Door A.

Here's where it differs from the Monty Hall problem: Instead of revealing a goat behind another door, a second person, independent of the game's host, randomly selects and opens one of the doors you didn't choose. Let’s say they open Door B, revealing a goat.

Now, you have the option to stick with your original choice (Door A) or switch to the remaining unopened door (Door C). The question is: Does switching improve your odds of winning the car?

The Counter-Intuitive Solution

The solution is, perhaps surprisingly, that switching does not improve your odds. This directly contrasts with the Monty Hall problem, where switching does double your chances.

The crucial difference lies in the randomness of the second person's door selection. In the Monty Hall problem, the host always reveals a goat, providing additional information that biases the probabilities. In the Bigloop Doors Paradox, the random selection of Door B by the second person doesn't offer that same informational advantage.

Let's break it down:

• Initial Choice: Your initial probability of selecting the car is 1/3, regardless of which door you choose.
• Random Door Opening: The act of opening Door B and revealing a goat doesn't alter the initial probability. There's still a 1/3 chance the car is behind Door A and a 2/3 chance it's behind either Door B or Door C. Since Door B is eliminated, that 2/3 probability now entirely rests on Door C.

Therefore, while it might seem like switching should improve your odds, it maintains a 1/3 chance. Sticking with your original choice also remains at 1/3.

The Importance of Conditional Probability

This paradox underscores the importance of understanding conditional probability. The seemingly random act of opening a door actually carries crucial information in the Monty Hall problem, making switching advantageous. However, in the Bigloop Doors Paradox, this additional information is absent due to the random nature of the second door opening. The prior probability distribution remains unchanged.

Conclusion

The Bigloop Doors Paradox serves as a powerful reminder that our intuitions about probability can be misleading. By carefully analyzing the conditions and applying the rules of conditional probability, we can unveil the surprising truth behind this seemingly simple yet complex puzzle. It highlights how a seemingly minor change in the setup can drastically alter the probabilities and challenge our assumptions.