You are exploring an ancient treasure chamber with 9 identical vaults. Hidden inside one vault is a priceless artifact, while the other 8 vaults contain nothing. Initially, the artifact is equally likely to be in any of the 9 vaults. You are allowed to select one vault to open, hoping to find the artifact. Before you open your chosen vault, your guide, who knows exactly which vault holds the artifact, opens 4 other vaults revealing them to be empty. The guide then offers you the choice to switch the vault to be opened to any of the remaining 4 vaults. Assuming the guide always opens 4 empty vaults and chooses them randomly from the possible empty ones, should you stick with your initial vault or switch? What is your probability of finding the artifact if you switch to one of the 4 remaining vaults?

This problem is an extension of the well-known **Monty Hall problem**, where switching vaults increases your probability of success. Let's break it down step by step to analyze whether switching is beneficial and what the probability of finding the artifact is if you switch.

### Initial Setup:
- There are 9 vaults.
- You don't know which one contains the artifact, so initially, the probability of the artifact being in any particular vault is \( \frac{1}{9} \) for each vault.
- You pick one vault (call it Vault 1), and before you open it, your guide, who knows which vault holds the artifact, opens 4 of the other vaults, revealing them to be empty.
- Now, you are left with 4 unopened vaults: your original choice (Vault 1) and 3 others.

### Two Possibilities:
- **Case 1**: The artifact is in the vault you initially chose (Vault 1). The probability of this happening is \( \frac{1}{9} \).
- **Case 2**: The artifact is in one of the other 8 vaults. The probability of this happening is \( \frac{8}{9} \).

Now, the guide has eliminated 4 vaults from the 8 other vaults. This means that if the artifact was in one of the 8 vaults you did not choose, it must be in one of the 3 unopened vaults remaining after the guide has opened 4 empty vaults.

### Probability Analysis:

1. **Sticking with your original choice (Vault 1)**:
   - The probability that the artifact is in the vault you initially chose is still \( \frac{1}{9} \), because no additional information has been revealed about your vault.

2. **Switching to one of the other 3 unopened vaults**:
   - The probability that the artifact is in one of the 8 vaults you did not initially choose was \( \frac{8}{9} \).
   - Since the guide has opened 4 empty vaults, the probability that the artifact is in one of the 3 remaining vaults is still \( \frac{8}{9} \), but now it is concentrated among the 3 unopened vaults. Thus, the probability of finding the artifact in one of these 3 vaults is \( \frac{8}{9} \) spread equally over the 3 vaults. So, the probability of the artifact being in any one of these 3 vaults is:
   \[
   \frac{8}{9} \div 3 = \frac{8}{27}
   \]
   for each of the 3 unopened vaults.

### Conclusion:
- If you stick with your original choice, the probability of finding the artifact remains \( \frac{1}{9} \).
- If you switch to one of the 3 remaining unopened vaults, your probability of success becomes \( \frac{8}{27} \) for each vault. Since there are 3 vaults to choose from, the combined probability of success if you switch is:
   \[
   3 \times \frac{8}{27} = \frac{24}{27} = \frac{8}{9}.
   \]

Thus, the probability of finding the artifact is much higher if you switch to one of the remaining unopened vaults. Therefore, **you should switch**, and the probability of finding the artifact if you switch is \( \frac{8}{9} \).

Would you like more details or have any questions?

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Here are 5 follow-up questions to expand your understanding:
1. What would happen to the probabilities if the number of vaults was increased to 12?
2. How does this problem compare to the original Monty Hall problem with 3 doors?
3. What if the guide revealed 6 vaults instead of 4? How would the probabilities change?
4. What is the probability that the artifact is in one of the 4 vaults opened by the guide?
5. How would the analysis change if the guide did not know where the artifact was?

**Tip**: Always reconsider probabilities when new information is revealed, as it can drastically change the outcomes!

Explore the probability problem involving 9 vaults and a priceless artifact. Learn whether you should switch after 4 empty vaults are revealed and calculate your chances of success using probability theory and Bayesian reasoning. Suitable for high school students in Grades 10-12.

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