The first number partitions the distribution. Unless the areas on either side of the partition are equal, there is a greater than 50 percent chance that the second number will be drawn from the larger partition. Assuming that the three numbers are independent and identically distributed, the probability of drawing the third number from the larger partition is the same as the probability of drawing the second number from the larger partition. Basically, the second number determines whether the third number will be larger or smaller than the first.
Shawn On Thu, Jun 9, 2011 at 9:09 AM, ERIC P. CHARLES <[email protected]> wrote: > Sarbajit, > Great point, but let me make it a bit more complicated. Possibilities marked > with a "+" indicate situations in which we will have a probabilistic > advantage in our guessing, possibilities marked with a "-" indicate > situations in which we will have a probabilistic disadvantage in our > guessing: > 1) A below B below > 1a) and A below B + > 1b) and B below A - > 2) A below B above + > 3) A above B below + > 4) A above B above > 4a) and A above B + > 4b) and B above A - > > Eric > > P.S. The case of a single bounded distribution is definitely the hardest for > me to think about, a double bounded or unbounded distribution seems much > more intuitive. Also, the restriction to guess relative to A makes it harder > for me to think about. Imagine instead that all we did was guess that the > third number would be above the smallest of the first two. > > On Thu, Jun 9, 2011 08:35 AM, Sarbajit Roy <[email protected]> wrote: > > 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. 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.) >> >> -- 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]> 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]> >>> 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]> >>>> 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 >> >> > > Eric Charles > > Professional Student and > Assistant Professor of Psychology > Penn State University > Altoona, PA 16601 > > > > ============================================================ > 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
