What are the odds for finding consecutive numbers?

I wasn't particularly good at statistics, so I thought I'd pose the question here. What are the odds of finding two watches with consecutive serial numbers? Having collected for a number of year one particular brand, I recently came to own such a pair. So I paid a bit more than the going rate for the second one... but I think it was worth it. But just how "rare" is this pair?

For the sake of the discussion, let's presume the company made 1,000,000 watches and that they are all still exist. In actuality, I would reason that about 20-30% are trashed.... but sticking with the round number of 1 million, what are the chances?

A preemptive begging of forgives from all my former math teachers.... I was too busy working on watches!

Probably the odds are fairly low, say there were 1 million watches made then each one million watches would have a companion serial number one up one down (except for the first and last). If you are looking for that situation I am sure you could end up with two serial number together depending on how long you look. It wouldn't happen every day but often enough that if you looked you could get two serial numbers together.

As an engineer, I look at it from a mathematical standpoint. I will try to explain it simply, in layman's terms, without getting into rigorous probability and statistics math.

Say you have a given movement number of a particular model. There are at most two other movement numbers that will form a consecutive pair with your watch, one higher and one lower. Now the probability of finding one of those two watches depends on the number of surviving examples.

This is consistent with Lin's theory that you have a better chance of finding a consecutive pair in a scarcer watch, as your two chances of success represent a larger percentage of the examples in existence. If there are 100 examples surviving, you have about a 2 out of 100 probability, versus a 2 out of 100,000 probability for 100,000 examples surviving. The probability of success is around 1000 times lower for a randomly selected watch of the more common type.

Wouldn't you think if there were only 100 samples available then ther could only be 99 possible consecutive matches whereas with 100,000 samples, consecutively numbered that is throughout the whole set, that there are 99,999 possible chances of finding consecutive numbers?

Are we excluding the possibility that the the two with consecutive numbers are not the same model watch? What about last of one run and the beginning of the next run but a different model?

Steve

About as common as seeing a flying pig?

I think we have seen a few such examples on this board over the last several years. I have 42880 and 42881 Ball Hamiltons and am pretty sure I have heard others mention their unusual pairings.

I will leave the math to someone else but knowing that only a small fraction of these seem to turn up, it is definitely a rare occurrence.

you will locate a post and pic of mine at

post and pics


happy hunting

Here is a LINK to my triplets.

And here they are in Sangamo Data Base on this site.

EXCLUSIVE FEATURE - ILLINOIS SANGAMO SPECIAL MODEL 9 DATABASE

Scroll to these three numbers in bold type for the description.

2868819
2868820
2868821

I think the odds are low but not mega-lotto low, first most people are not looking for consecutive serial numbers but the ones that are find them, sometimes even 3 in a row. I also agree the more one model is made the better your chances since I doubt that one person locks into one serial number but will get two close and find a matching one for the first or second. Also if have 4 or 5 of the same model your chances increase.

I had to be sad and work out the maths (I think). I'd agree that if all of the watches survived then the chance of any two being a pair is about 2 in a a million (If a million were made), but we see more than one of each watch. Assuming we saw 100 I think the chance of any two being a pair is roughly 20,000 of a million, or about one in 50, if I've done my maths right. This is only if all of the million survived, and in reality this is unlikely so the chances are lower, but it could still happen. I noticed Jon Hart has a Hamilton 936 with the consecutive number to mine, although it's even less likely I'd have ever seen it over in this country.
-Joe

This may be a stupid question, but here goes, suppose a jeweler orders 4 or 5 Hamilton 992 or Bunn Specials, B W Raymond's,or whatever, all of the same model, grade, jewel count, etc. Would the watch company have supplied random serial number watches or consecutive numbers. Also, was the serial number applied to the movement upon being finished at the factory or during assembly, and were movements numbered consecutively upon completion.

I said I wasn't going to do the math but the statistician in me can't resist.

Assuming a 10 percent survival rate.

Each run of a hundred would have 10 survivors.

the first watch could have any number.

the second watch has two chances in 99 of matching the first watch.

The third watch has two chances in 98 of matching the first and two chances in 98 of matching the 2nd.
the fourth watch has two chances in 97 of matching the first watch and two chances in 97 of matching the 2nd watch and 2 chances in 97 of matching the third.
Fifth watch has 2 chances in 96 4x
6th watch has 2 chances in 95 5x
7th watch has 2 chances in 94 6x
8th watch has 2 chances in 93 7x
9th watch has 2 chances in 92 8X
10th watch has 2 chances in 91 9x

That works out to 90 chances in an average of 95 events that there will be a consecutive match amongst the 10 survivors. A high probability that two consecutive numbers will survive in each run of a hundred.

Understand that this is an oversimplification of the statistical chances. Even on the 10th watch which had 18 chances in 91 of being a match, that single event had less than a 20 percent chance of being a consecutive match and just as in flipping a coin where you would expect to hit heads fifty percent of the time, that does not make it a certainty that 10 flips couldn't all be heads, just unlikely.

You could have a fifty percent survival rate and all the survivors could be non consecutive>
ie: # 1 3 5 7 9 11.............. but the odds on that are just astronomical!

A higher survival rate just means that it is even more likely that consecutive pairs still exist and are waiting out there to be noticed.

Keep in mind that the more desirable the watch the more likely it is that it will come to the watch collectors attention. Most of the survivors of common watches will remain anonymous as no one really cares.

Now the chances that one owner will acquire the pair are quite a bit more remote.

happy hunting

Those look like Elgin hands-??