A lucid analysis. BUT, If we consider the median = 1/2 infinity case, we end up with 3 "equally probable" cases. a) both number below median b) both numbers above median c) one below and one above median
alternatively we could get 4 "equally probable" cases 1) A below B below 2) A below B above 3) A above B below 4) A above B above I'm still unable to see how we get a "better than 50%" edge by knowing the 2nd number. The "normal" distribution would not apply to random numbers - which are evenly distributed ie. "flat". Sarbajit On Thu, Jun 9, 2011 at 5:46 PM, ERIC P. CHARLES <[email protected]> wrote: > Ok, I'm a bad person for not reading the cited paper, but I was thinking > about problem late last night. I keep thinking that we need to make > assumptions about the distribution (regarding bounds and shape), but then I > can't figure out a combination of assumptions that really seems necessary. > This is because any distribution has a median (even if it is an incalculable > median, like 1/2 infinity). Using that as the key: > > Given two randomly generated numbers, odds are that one of them is above > the median, the other is below the median. We need two numbers, so that we > can tell which one is which. If we restrict ourselves to making a guess > relative to the first number (because that's what I think Russ was saying), > then when the first number is the smaller one, we guess that it is below the > median (and hence the third number has more that a 50% chance of being above > it). Reverse if the first number is the larger one. > > Of course, sometimes we are wrong, and both random numbers are on the same > side of the median... but on average we are still better off guessing in > this manner. If we know the shape of the distribution, it should be pretty > easy to calculate the advantage. For example, if the distribution is normal, > the smaller score will (on average) be one standard deviation below the > mean, and hence 84% of the distribution will be above it. > > Eric > > > On Wed, Jun 8, 2011 11:10 PM, *Russ Abbott <[email protected]>* wrote: > > It doesn't establish the range. All that's really necessary is that there > be a non-zero probability that the second number falls between the first and > the third. On those occasions when it does you will have the right answer. > On all others you will be right 50% of the time. I saw it in a reprint of > this > paper<http://www.americanscientist.org/issues/issue.aspx?id=5783&y=0&no=&content=true&page=2&css=print>. > Look for David Blackwell. > > What I like about this phenomenon is that it feels like action at a > (mathematical) distance -- similar to the Monte Hall problem in which > showing the content of one door makes it better to switch choices. (If you > don't know this problem, it's worth looking up, e.g., > here<http://www.askamathematician.com/?p=787> > .) > > *-- Russ Abbott* > *_____________________________________________* > *** Professor, Computer Science* > * California State University, Los Angeles* > > * Google voice: 747-*999-5105 > * blog: *http://russabbott.blogspot.com/ > vita: http://sites.google.com/site/russabbott/ > *_____________________________________________* > > > > On Wed, Jun 8, 2011 at 6:52 PM, Sarbajit Roy > <[email protected]<#13074553f57f0e8b_> > > wrote: > >> How does knowing the second number establish the range ? Is there any work >> on this. >> >> Sarbajit >> >> >> On Thu, Jun 9, 2011 at 1:15 AM, Russ Abbott >> <[email protected]<#13074553f57f0e8b_> >> > wrote: >> >>> Russell Standish has the right idea. If you knew the range, say the >>> first number is higher/lower than the third depending >>> on whether the first numbers is greater than or less than the middle of the >>> range. Since you don't know the range, the second random number is used >>> instead. Say higher/lower depending on whether the first number is >>> higher/lower than the second. >>> >>> Also, think about it this way. If the middle number is either greater >>> than or less than both the first and the third, you have a 50% chance of >>> being right. If it's between the first and the third, the strategy described >>> will always be right. Presumably there is a non-zero probability that the >>> second number will be between the first and the third. Therefore one has a >>> greater than 50% chance of being right. >>> >>> *-- Russ * >>> >>> >>> On Wed, Jun 8, 2011 at 10:06 AM, Owen Densmore >>> <[email protected]<#13074553f57f0e8b_> >>> > wrote: >>> >>>> Do you have a pointer to an explanation? >>>> >>>> -- Owen >>>> >>>> On Jun 8, 2011, at 12:11 AM, Russ Abbott wrote: >>>> >>>> > Although this isn't new, I just came across it (perhaps again) and was >>>> so enchanted that I wanted to share it. >>>> > >>>> > Generate but don't look at three random numbers. (Have someone ensure >>>> that they are distinct. There is no constraint on the range.) Look at the >>>> first two. You are now able to guess with a better than 50% chance of being >>>> right whether the first number is larger than the unseen third. >>>> > >>>> > I like this almost as much as the Monte Hall problem. >>>> > >>>> > -- Russ >>>> > >>>> > ============================================================ >>>> > FRIAM Applied Complexity Group listserv >>>> > Meets Fridays 9a-11:30 at cafe at St. John's College >>>> > lectures, archives, unsubscribe, maps at http://www.friam.org >>>> >>>> >>>> ============================================================ >>>> FRIAM Applied Complexity Group listserv >>>> Meets Fridays 9a-11:30 at cafe at St. John's College >>>> lectures, archives, unsubscribe, maps at http://www.friam.org >>>> >>> >>> >>> ============================================================ >>> FRIAM Applied Complexity Group listserv >>> Meets Fridays 9a-11:30 at cafe at St. John's College >>> lectures, archives, unsubscribe, maps at http://www.friam.org >>> >> >> > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > > Eric Charles > > Professional Student and > Assistant Professor of Psychology > Penn State University > Altoona, PA 16601 > > >
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