Ming Roulette

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Interviews at hi-tech companies often include puzzles. Here is one that has made the rounds in spite of its violent content.

You are an antique dealer and I am an unsavory character who has lent you money. Instead of the usual baseball bat, I have brought a revolver to emphasize the importance of repaying me. I put two bullets in adjacent chambers of the six-shot revolver and spin the chambers. I point it at a valuable vase and pull the trigger.

Click.

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No bullet has fired. The chamber has advanced by one. I am prepared to try again.

Warm-Up:
Is it better for the vase if I spin again or not before pulling the trigger a second time?

Solution to Warm-Up:
It's better to keep the chamber where it is, i.e., not to spin. Let E represent an empty chamber and B represent a chamber with a bullet.

Putting the chamber first used at the beginning on the left, the gun could have been in any of the following configurations with equal probability:

BBEEEE
EBBEEE
EEBBEE
EEEBBE
EEEEBB
BEEEEB

Because the vase wasn't broken, the equi-probable configurations remaining are:

EBBEEE
EEBBEE
EEEBBE
EEEEBB

If I don't spin, then the vase is broken in only 1 in 4 cases on the second trigger pull. If I spin, the vase will be shot with a 1 out of 3 chance. So, you don't want me to spin.

End of Warm-Up

Now suppose I am going to pull the trigger two more times, for a total of four. If the vase survives the second trigger pull, then the revolver must have started in one of these configurations:

EEBBEE
EEEBBE
EEEEBB

1. Do you ask for a spin of the chamber before the third trigger pull? How about before the fourth trigger pull? Using your best strategy, what are your chances that the vase survives given that it has survived the first trigger pull?

2. How would your strategy change if there had just been one bullet in the chamber from the beginning?

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