Problem #420

Expected Universe I
Public 07/31/17 6xp Probability 30.4%

Initially the Universe does not exist. The God (or whatever you name it) sits idle within nothingness. Luckily the God has a little friend that keeps asking questions every day but just one for each day.

The little asker has 3 questions of the form:
(1) Will you say YES to my next question?
(2) Will you say NO to my next question?
(3) Does the Universe exist now?

(a) God can only answer as YES or NO. God always answers immediately within the same day.
(b) The answer to (3) is always deterministic, namely there is only one choice always, in default case being always NO, but can only be forced to YES through a sequence of (1)s and (2)s
(c) If (1) or (2) asked right after (3) or at start, the answer is nondeterministic, namely God can answer at will as YES or NO. This is the only case for God to be nondeterministic.
(d) A sequence of (1)s and (2)s which goes on after the preceeding (1) or (2) is deterministic, and God has to answer possible successive (1)s and (2)s accordingly and never cheat.
(e) If a (3) appears at the end of a sequence of (1)s and (2)s, the answer is obviously deterministic. But as noted in (c), a (1) or (2) appearing after a (3) or (3)s, will again cause a nondeterministic answer from God.
(f) These rules go on until the God is forced to answer as YES to Universe existence question, and the Universe finally emerges.

Here is a sample run to get familiarized with the concept:

A: Does the Universe exist now?
G: NO (initially it does not exist!)
A: Will you say YES to my next question?
G: NO (nondeterministic, can say either...)
A: Will you say YES to my next question?
G: NO (deterministic, has to say NO)
A: Will you say NO to my next question?
G: NO (deterministic)
A: Will you say NO to my next question?
G: YES (deterministic)
A: Does the Universe exist now?
G: NO (deterministic!)
A: Will you say YES to my next question?
G: YES (nondeterministic as form (3) breaks determinism)
A: Does the Universe exist now?
G: YES (finally!)

So the asker's choices are uniformly random, and also the God's will (when possible) is uniformly random. With these final assumptions, what is the expected number of days needed for the Universe to emerge? Round your answer to 3 decimal places.



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