June 2nd, 2007
Feel free to post solutions (or solution related comments) to this page!
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November 8th, 2007 at 1:13 pm
I draw a random number X with a probability positive over R. If X is smaller than the number I have on the envelope, I will say that it’s because it’s the larger, if not I will say it is the smaller. I will be right 1/2+a/2 of the times, with a being the probability that the number X is between your two envelopes.
November 8th, 2007 at 6:00 pm
Well, I think there is a slight imprecision in the statement: “with a probability positive over R”. What you actually want to do is generate a number that has a positive probability of falling between every 2 distinct positive integers (for example, let y be geometrically distributed with any parameter p between 0 and 1, and let x be y + 1/2).
This solution easily extends to the case where what is written in the envelopes are not integers but elements of some arbitrary ordered set, on which you can define a probability distribution that gives a positive probability to a dense subset (dense in the sence that between every two elements of the original set there is an element of the subset – note that with this wording, Z+1/2 is dense in Z).
Another solution to the riddle: say that the number x you are seeing is the lesser of the two with a probability of 1/2+1/x (or some other formula of the sort).
November 8th, 2007 at 6:21 pm
When I hear number I think “real”, not integer! so my solution works even if the two numbers are reals (even negative ones). It will therefore also work for N, Z, Q and other subsets of R.
November 8th, 2007 at 7:01 pm
Got you (I shouldn’t have used the term “number”, my bad
).
And, yeah, generalizing the riddle to the reals is possible (that is the second part of the first paragraph of my comment). My point was that the statement “a random number X with a probability positive over R” should be replaced by “a random number X with a positive probability of being between every two distinct elements of R” (this is probably what you meant).
November 8th, 2007 at 7:06 pm
For me R = IR = the real numbers. So “a random number X with a positive probability of being between every two distinct elements of R” is just a continuous random number (say, a normal distribution). This is easier to imagine than a random distribution on Q!