bushindo

You people are making this too complicated =) Let's say that a speaker wants to convey the statement "I want dinner now". If the speaker is type B, he should ask "Is the sentence 'I want dinner now' grammatically incorrect?" If the speaker is type A, he should ask "Is the sentence "I want dinner now' grammatically correct?" There might be some work getting all the denizens familiarized with this convention, but the bonus is that once established, the convention does not require a speaker to first identify whether he is type A or type B.

Here's how I believe the prisoners should approach this

But it is easy to devolve into a Wine-in-Front-of-Me situation

My take on this

Let me restate the proof in a more direct way, and we'll see if we get anywhere

How about this proof then,

Let's relax the definition of monotone a bit and say that in a increasing monotone sequence, each number is bigger than or equal to than the last number. Same with decreasing monotone sequence. Then the statement in the OP is an extension of a previous puzzle

Great analysis, and I agree with your answer. Good solve.

I get the same results with a right cylinder as with the sphere

Here's the integral I used for Case 1, I like bonanova's approach much better though.

Assuming the arrival time for the two friends are independently, identically, and uniformly distributed over the hour, then

This puzzle statement may be undercontrained,

Aye, Captain,

Great solve. It is indeed surprising how efficient bonanova's and your algorithm is.

Some clarification please. What happens when the player removes a ball labeled with the number 1?

Now that bonanova and EventHorizon have posted solutions to the original problem, allow me to pose an extension. What is the average number of coin flip required for bonanova's and EventHorizon's approach to sampling the probability p? Extra bonus for putting the answer in closed form (i.e., without requiring an infinite sum).

You're right in that your scheme, which calls for consecutive flips until the outcome differs from the binary representation, returns a probability that is precisely p. I made a mistake earlier in saying that my scheme (flip the coin for a fixed number of time N) is 'equivalent', which it assuredly is not. While that scheme may be more intuitive, it sufferers from approximation error due to the fixed finite N, plus the fact that it is not as computationally efficient. Good point!

Thanks for the clarification. If that's the case,

By 'ideal distribution of elevator position', do you mean that 1) You wish to minimize expected the number of floors it would take for an elevator to reach a caller whenever he/she presses the elevator button? 2) Or do you wish to minimize the expected total distance that an elevator would travel when requested (i.e., caller A request the elevator from floor i to go to floor j; the nearest elevator travels from its idle floor to floor i, takes the caller to floor j, and then return to its idle floor)? Also, as it is, the problem is under-constrained since it leaves out an important piece of information, which is the probability of calls from the ground floor (or floor 0). Consider the following two scenarios A) Nobody in the building travels to the ground floor. All calls from floor 1 to 12 are equally likely, and each elevator trip travels to floor 1 to 12 with equal probability. In this case, probability of calls from the ground is 0. B) All building residents only travel from their floor to the ground, and from the ground to the their floor. Assuming that there is no net gain or loss in the number of residents, then probability of calls from ground floor is 1/2. Both scenarios A and B satisfy the OP, but they have different implication on the resulting `optimal' idle floor. Some clarification would be appreciated.

I was incorrect in the post above. Turns out we don't really need the prior distribution of the red balls. In the words of CaptainEd, "Durn, hit by Bayes again" :-)

I think the crux of the puzzle lies in the part that is highlighted red above. I assume that by 'undetermined number of red balls', you mean that the precise number of red balls is unknown. However, the solution to the puzzle requires that we assume something (a priori information) about the generating distribution of the red balls (e.g., poisson distribution, exponential distribution, gamma distribution, etc.). We could also start going into improper priors, but I'd rather not =). This puzzle seems dangerously close to a subjective argument about what type of prior distribution does 'undetermined number of red balls' mean, maybe the OP can clarify that a bit?

Great puzzle. I'd just note that there's equivalent way to do this that may be more intuitive for some people.

Congratulations, you got it. Great solve.

Question regarding this approach

All correct there, except for one detail The emperor would also like to request more details on this drinking scheme for the sake of completeness. You see, back in college the emperor majored in Political Science, which does not require any course in Computer Science, so concrete details on how to go about doing it would be very much appreciated.