Dark room problem, with a twist

[lorinda]

Hi there,
many of you will have recieved a number from me via internal mail, I hope this hasn't upset any of you :O

Anyway, the dark room problem was interesting enough to make me want to try it, but then I thought of a twist, what if you could talk to some of the people around you?
How quickly could a decent estimate be established?
Anyway, for those who are interested, the numbers have been sent out for an imaginary dark room, I took great care in randomly selecting the number of people who are in the room, and also promised myself that whatever numbers had to be sent out (even if they came out in a strange manner) would be the numbers sent.
Anyone who would like a number who hasnt got one, then please ask for one and i shall generate one for you via internal mail.
I would ask each poster to post a guess (The first people have it tough) as to the number of people in your dark room.
I think this could be a fun, and mathematically intersting exercise.
Basically, early posts should contain your number, and a guess (with any reasoning you like) as to the highest number possible (the number of people in the dark room)
I will refrain from posting the actual number for some time, and may even give people a 2nd number each if we only get a small sample reply.
Hope this is going to be an interesting and fun challenge, and I have kept my personal promise too, the numbers are random as is realistically possible

[lorinda]

Just to clarify,I obviously havent mailed x people (where x is the number in the room) with this message, but it can be assumed that I have,you have all been assigned a number from 1 to x, and all people wanting a number will get the same.

[lorinda]

Ok,something else i forgot to mention (after ages using internal mail and generating numbers i forgot my original draft for the post), I have a master copy of all the numbers I have sent so far, if I correct anyone, it wont be (most likely) because they are lying, but more likely that I sent them the wrong number,I sent many numbers out and it is possible I made typing errors, which would defeat the whole point of the exercise

[Dynasty]

I've got my number. I just don't understand the problem. It doesn't seem to be clearly defined. In addition, the number sent to me isn't useful if I don't understand the problem.

the dark room problem was interesting enough to make me want to try it

I'm already confused. You seem to be assuming that I and others already know what the "dark room problem" is. Was this discussed in another thread which I didn't see?

How quickly could a decent estimate be established?

An estimate of what? From information below, it seems we are trying to guess the number of people in a room.

I would ask each poster to post a guess (The first people have it tough) as to the number of people in your dark room.

In my dark room? Is there more than one dark room?

Let me restate the problem as I understand it. Please correct me if I haven't got it right.

You are in a dark room with an unknown number of other people. The only information you have is a number assigned to you. Everbody else also has a number assigned to them from 1 to x with x being the number of people in the room. By talking to others in the room, conclude how many people are in the room

I have #109. Therefore, there must be at least 109 people in the room. I have no other information beyond that. Therefore, I will guess that I have been assigned the median number and that there are 218 people in the room until further information becomes available.

[lorinda]

It was based on a post some weeks ago, but your wording is perfect (in my own current baffled state at least)

Something else to keep an eye on is the confidence levels, how close would you think the current guess (218) is, reasons behind that and so on.

I shall now keep out of this thread unless I have dropped any more clangers, or think of any interesting questions to pose.

Remember, if you want a number, just internal mail me.

[irchans]

My number is revealed at the end of this post so don't scroll to the bottom unless you want to see it.


Suppose you speak to n people and the highest number among those n is m. I suggest 3 methods for guessing the number (one silly, one simple and good, one overly complex but mathematically fun).

Method 1: (Maximum Likelihood) Just guess m.

Method 2: (Reverse? Least Squares?) Just guess m* (n+1)/n.

Method 3: (Least Squares using BruceZ's model) Assume you have some upper limit u on the highest number that lorinda knows /forums/images/icons/laugh.gif )

guess ( f(u, n-1) - f(m, n-1) ) / (f(u, n) - f(m, n) )

where f(x, n) = Integral[ 1/(y^n), {y, 1, x}]

( f(x, 0) = (x-1), f(x,1) = Log[x], f(x,2) = 1-1/x, ...)

So my guesses are below

































My number was 494. Originally, my guesses by the above methods were 494, 988, and (10000 - 494)/(Log[10000.] - Log[494]) = 3160. I like method #2 best --- 988. Then I saw Dynasty's post of #109 (and her guess using the median!). After seeing the 109, my guesses would then be 494, 741, and (Log[10000] - Log[494])/(1/494 - 1/10000.) = 1563. I like 741 best now.

(Any mistakes in this post are hereby blamed on my cat toots. Who is DEMANDING attension at this moment. /forums/images/icons/smile.gif )

[Mike Haven]

my number is 703

as i haven't met anyone with a higher number i will guess there are 703 people in the room

[irchans]

With Mike's record breaking 703, the survey results are now 109, 494, and 703. I now think the biggest lurker is 937 or 1313, but I guess the most likely result is that Mike is biggest. (Note: these guesses are based on the idea that people's motivation to share is independent of size.)

Note to people who are holding back: Large ones are best, but even if yours is small, be willing to share! /forums/images/icons/wink.gif

Feel free to quote me out of context should I run for political office.

[Mike Haven]

one scenario:

if i can only ask "what number are you?":

to have an evens chance of guessing the right number of people in the room you have to ask half the number of people of the largest number you have been answered

i am number 703

if i ask 351 other people and do not hear a higher number it is 50%/50% that i was the last man walking in

if i hear, say, 800 during my questioning then i have to ask 400 people, and so on, for it to be evens that i have spoken to the last man

[Mike Haven]

ok lolly

you win

i must know

was i the last man walking?

[baggins]

i was 539. doesn't help much.

[lorinda]

Oh but baggins..it helps confidence levels, and also ties down a more accurate estimate /forums/images/icons/smile.gif if only more of you would come forward.....