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Statistics Question: How Old is this Bar?
What you’re looking at is a bar made by arranging roughly a hundred bucks* in pennies over the surface and coating them in plastic. I can read the dates off of some of the pennies (those that aren’t flipped upside down), but quite obviously not all the pennies were minted in the same year.
Here’s the question: Judging solely by the dates these pennies were minted what year was this bar constructed? How many dates would I** need to read to have a reasonable confidence in that answer? Should I bother taking dates off of the dull pennies, or only focus on the shiny new ones?
*I arrive at this number by a Fermi estimation. Assuming the bar is twenty feet long, two feet across, and that the radius of a penny is 9.775 mm (thanks Bing) how much is it worth?
**Well, not me personally but I do have a research team nearby. I doubt I’m welcome back in that establishment after telling the bartender that she was going to have trouble finding her tip.
Published in Science & Technology
Yeah. If in a sample of
3231 (damn Canadian funny money) I had twelve pennies from specifically 1997-1999 that would have been worth mentioning.I plan to do a sample myself, but as I no longer use coins, and only have the odd trips to update my coin stack, I cannot really address the leading edge. But the tail will be unaffected.
Thinking of getting some from the store. But they’ll be loth to part with meaningful quantities. Hmm.
If I recall correctly, this place first opened its doors in either 2006 or 2007. It had also been renovated several years prior as well.
You got all that from the coins?
I’ve had a hunch about this and I finally put my finger on it. I suspect that this is a pessimistic take, that it overstates the number of coins required for various orders of equation (that is, for various steepness of the drop-off with age), particularly as the order gets higher. This is of course continuous modelling of a discrete problem.
CLEARLY, the answer for the uniform case is probably correct.:
Chance of NOT drawing 1 in 100 is 99 in 100, or .99.
.99^114 is the first integer power to get below .10
That is a ten percent of chance of failing to get even a single (selected) year coin even after 113 coins. That’s the same as a 90% chance of succeeding. Since we are answering a question based (solely) on the information gained this way, and since I am (so far) ignoring fencepost effects about [current year], this 90% is also the CI.
Yet, CLEARLY this model must be wrong about tenth-order models. To whit, we cannot even construct a tenth-order model for 100,000 coins, except to stack them all up in current year and ignore the fact that we are still a dozen orders of magnitude short. This means that to select any coin out of 100,000 would be guaranteed to get [current year].
So the model may be good for problems with ridiculously large numbers of coins (we’re going to need 18 zeroes, minimum), but this is not that problem.
I think the model performs perfectly if uniform, adequately for first order, shabbily for second order, and I wouldn’t put money on the rest no matter what.
I do know that given the actual distribution of coins, we can give an age with a very narrow CI. This has been an exercise in trying to model the distribution as if it were continuous — it is not.
This just gets worse and worse.
Actual US Mint penny production figures are a flat-out mess. This complicates any attempt to generalize the actual portion of any given year’s pennies in the bartop. We would really need to start with penny production figures and then (I presume, and where I was going with the whole nth-order models) exponentially decay them into the past. Again, this ignores leading-edge effects until later.
However, that should give us a suitable nominal population of pennies by year under the bartop, including the all-important expected count of current year pennies (current when the bar was built).
The same problem will also mask expected variation in attempts plain-old count coins from the store, etc. In a large enough penny sack, there will be large swings which do not go away — grab another sack and the supposedly random variation will show the same sequence — because that’s really how pennies are produced.
Miserable for modelling, but data is better than not-data. Ugh.
This post and thread is why I love Ricochet and all of you commenters out there.
And I’m really happy to have met some of you in person.
The high school my daughters attended had a tradition of the Senior Prank. A group farm kids released some pigs into the hallways in the middle of the day. The custodial staff and the office staff quickly rounded up three of them, but spent the rest of the day searching for one more. What they did not know is the the kids painted the numbers 1, 2, and 4 on the three pigs they had released.
[Deleted because several others had made similar comments]
Or AOC
Your chances improve if all your purchases are done with cash.
Lots of peeping Abes.
I have multiplied matrices and done other matrix math, though not recently. I don’t know about Markov chains, but it sounds interesting.
People pay a lot of money to go and see and photograph whale flukes. They are not easily dismissed.
There really is only one answer. Everyone overcomplicates it. Take the latest date of the shiniest pennies. It’s between then and now. That’s the only certainty. A dirty penny won’t be new. Nobody mentioned that 50% of the pennies are upside down.
Now that this post is thoroughly beaten to death, here is my bar top. 3/4″ epoxy over curly maple. The knot took a full quart to fill and I tucked a fake diamond in it. It took 2 gallons total.
You need to look at them all for certainty, or to specify confidence limits. A single coin from 2014 invalidates everything before that year. Ditto for 2015, etc. And since the bar was likely not constructed on December 31, the year of its fabrication will probably be under-represented, too. If you are lucky and see a 2022, then case closed.
There is likely only one answer, and that is not it.
So survey all 10,000 pennies (you have to look at one to determine that it is upside-down, or shiny), and then take the latest date, and admit only that the date of construction is sometime between then and [now]. This is a very imprecise answer with absolute confidence, and conducted via the most expensive sampling method — a full survey.
Even with that methodology, how would you take that method and actually answer the question asked:
You have answered a different question. If you are averse to using statistical methods to answer statistical questions, I don’t know what to tell you.
Finally, why do you think that 50% of the pennies are upside-down? Have you surveyed every penny? You cannot be certain of that otherwise. Perhaps you are more comfortable with statistics than you would like to think.
The post is not beaten to death. We’re identifying issues. Just getting started!
Nice bar top in your photos!
I be wrong about under-representation of the current year. I just pulled 20 pennies from my coin jar. From the top, thus the most recent additions.
1983, 1985
1993, 94, 97, 97
2017
2 x 2019
5 x 2021
6 x 2022
The last eleven I can understand, they are fresh from the mint to the bank to the grocery store. But what of the others? What happened to 1998-2016?
Randomness…
Not true.
If you sample all but one, what are the odds that the last one will be after all the others? Very small indeed. This will be a trivial uncertainty to factor into the existing uncertainty alluded to by Chowderhead. As the number of unsampled coins grows, the uncertainty grows as well. If it didn’t, then looking at one coin would be just as good as looking at all but one coin.
So, if we are splitting hairs here how about this? You cut every penny out of the epoxy. Log the date of every penny. Now you need a certificate of authenticity to determine if the latest penny is marked correctly and not a fake. All anyone knows for certain is it was made before the post was made on January 1’st, 2023, and after two part epoxy (or glass) was available.
The debate nobody acknowledges is how much probability and uncertainty are you willing to accept as fact.
One coin can be just as good as all if the one coin is from the current year.
Yeah, I agree that not all CIs are made up.
In pure statistics, they are not made up at all, though in pure statistics, we assume the applicable probability distribution function (which we called a “pdf,” I think, back when it was the only thing that I knew of called a “pdf”).
In application, we don’t usually know the distribution function, though we can know in some situations like cards or dice rolls.
Have you even read this thread? People discussing CIs? Arguments about the meaning of the answer? And lots of people attacking the age of the bar itself, rather than “how many coins for reasonable confidence”?
I offered a 90% CI as a “reasonable confidence,” and promptly offered a wildly optimistic first stab. Been refining that.
Funny of you to complain about “splitting hairs” when you’ve chosen a peculiar extreme to stand on — absolute confidence in zero knowledge after a certain date. This answers the question in the same sense that a point is a circle.
Or we could always ask the owner when it was put in.
Heh.
Good catch of an edge case (and others have spotted it). I am proceeding as if the actual date of construction is not so recent as to give itself away by running into that edge, or to overwhelm other probabilities.
I moved on approximately between comment 50 and 100. I thought this was just statistics and probability fun. I didn’t know it was such a touchy subject.