Just look at this one...

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1 in 10 Million for this combo. A totally random number, so there would be no hope in digging this out of a regualr sub collection (such as radars). The scarcity?  Especially in near consecutive prefixes of the same denomination?
What do you guys think?

http://www.members.shaw.ca/collectorsgallery/CGPaperJourney.html

The two matching One dollar bills.  (PS I would have posted the pic, but the directory is too full of pics it said.)

Here's the picture:
{http://www.members.shaw.ca/collectorsgallery/CGPaperJourney_files/image010.jpg}

Sudzee had a few sets like this for sale not long ago with 5's and 10's. I think he was asking $20 over face value for his sets.

These probably aren't as hard to put together as you'd think. For example, out of 3000 randomly selected notes you have 4498500 pairs of notes which gives you a 36% chance of finding at least one matched pair.

What I'd REALLY like to see would be a 1973 $1 and a 2004 $20 with EXACTLY matching serial numbers.  The AL_ prefixes would allow for that.  Has anyone ever seen such a thing?  I have asked local dealers if they ever have and they just laugh and tell me "good luck."

These probably aren't as hard to put together as you'd think. For example, out of 3000 randomly selected notes you have 4498500 pairs of notes which gives you a 36% chance of finding at least one matched pair.

Here is where I beg to differ.
You can start with any old note.  1/1, no big deal.
Your second note though, must be the exact match, so out of a run of 10 million less a few, your odds become 1/10M (est.)

I don't understand the 3000 randomly selected notes bit- could you elaborate?  

Sudzee had a few sets like this for sale not long ago with 5's and 10's. I think he was asking $20 over face value for his sets.

I am working on a small article about this kind of thing.  It has been discussed.... IMO $20 over face is WAY below what they should be, given the effort to assemble, and the odds of them coming together.   I have seen a pair for $80 on a dealer website.  That is alittle more in line IMHO.


Thx
Huds

And as far as the $20 and the $1, now I have a new project.
Step 1: record every single living AL_ prefix $1 that exists.  
Step 2: get into the $20s before this chance runs out.

For example, out of 3000 randomly selected notes you have 4498500 pairs of notes which gives you a 36% chance of finding at least one matched pair.

So are you saying if I had 10,000 notes I have a 99% chance of finding a match? :-? :-? :-?

So we will just have to leave that to the master "H" lol

Arthur

Here is where I beg to differ.
You can start with any old note.  1/1, no big deal.
Your second note though, must be the exact match, so out of a run of 10 million less a few, your odds become 1/10M (est.)

I don't understand the 3000 randomly selected notes bit- could you elaborate?  

Out of a sample of 3000 notes, you have 3000 choices for the first note in your pair and then 2999 remaining choices for your second note.
But, there are two different ways to assemble each pair, that is, you could have chosen one of the notes first, or the other. So the total number of pairs possible is 3000*2999/2 = 4,498,500.

Now, the likelyhood that any given pair is NOT a match is 0.9999999   (9,999,999/ 10,000,000).

The likelyhood that NONE of your 4,498,500 pairs are a match would then be 0.9999999^4,498,500 = 0.6377

So the likelyhood of finding a matched pair is 1-0.6377 = 36.33%

So are you saying if I had 10,000 notes I have a 99% chance of finding a match? :-? :-? :-?

Correct. Same calculations with 10,000 notes gives you 49,995,000 pairs.
0.9999999^49,995,000 = 0.00674
or 99.326% chance of finding at least one match.
Chances are you'd find several (~5) matched pairs with this many notes to choose from.

You could even further increase your odds by restricting your sample to meet some arbitrary but easily met criteria, such as only notes with a serial number ending in "1". But the key is having a large sample to choose from since your chances of finding a match increase with the SQUARE of the sample size.

You also have to keep in mind that if you're choosing notes from a small set of prefixes, the difficulty increases slightly since, of course, any pair from the same prefix has a ZERO likelyhood of being a match (unless you're talking about AL_ $1 & $20...).

Ok, enough with that. I should be doing my math homework right now.  

3000 choices for the first note in your pair and then 2999 remaining choices for your second note

What I don't understand is: of these 2999 notes, how can the fact that they are of 10 Million possible combinations be ignored?

I want to break this down to see exactly where I my logic is incorrect please. (Maybe I will use this in a math lesson plan)
1. I basically have 10 Million choices for the first note, out of 10 million. Reducing that down becomes 1/1.
2. Any particular number will have the same odds of being found:  1 in 10 Million.
3. It doesn't matter which note you pick first, the second note will exhibit the probability of point #2.
4. Combining the probabilities, you get
1  x  1     =  1
1     10M      10M

5. Much like two decks of cards:
To pick any first card is 1/104
For your second card to be the exact match is 1/103.  
Total probability: (1/104) * (1/103) = whatever it is.
Is this true or is this false?   :-?

6.  Think of a die as a prefix series with 6 notes total.
Roll any number on a die.  (1/1 because any side is okay)
Roll the same number on the next die (1/6)
Probability of rolling the same on both dice is 1/1 * 1/6, isn't it?

If I am going to be teaching math, I better hit the books if this isn't right. lol  I just don't understand where this is refuted by statistics and samples.  Please enlighted!! :-/

Hudson, for any pair, the chance of having a match is 1/10M like you said.

The point is, if you have a large sample of notes to choose from, you end up with a HUGE number of possible pairs, any one of which has a minute chance of being a match.

If you have a Million+ possible pairs and each has a likelyhood of 1/10M to be a match, then your chances of finding one start to become somewhat realistic.

Perhaps if all the forum members got together and made one list of all their serial numbers, we'd find many, many matched pairs...  

h_p

Yikes. I guess I need to carry a stock of 10 Million notes then if I want to find another pair.  
Huds

I agree with happy's conclusion (although I got 36.23% for 3,000 notes using Excel when I did the calc.)  Another way of expressing the probability for finding a match in a group of n notes is:

nPr(10,000,000 ; n) / 10,000,000^n,
or alternatively,
10,000,000! / [(10,000,000 - n)! * 10,000,000^n]

This formula slightly over estimates the probability because it assumes each serial number is independent.  (This is not true because if you have ABC123456789, then there can be no other such number with prefix ABC.)  However, the effect of this on our estimate is very small.

Very few math programs will be able to calculate the above for n>50.

Brad

wow

This excites me when I see money and math mix together...  

They go hand in hand...