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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
I’m going to guess it was made in 2017. Five years ago, now six, poured epoxy resign bar tops were very popular.
A judge I met noted that her husband was an interior designer for commercial spaces, and in almost any public space they went to he could identify within a six month range when the space had last been remodeled based on the design elements used in the space.
Why remodel? I just like it when the trends to come back around to my point of view.
This isn’t A/B testing.
This is a question for Marilyn Vos Savant . . .
Yes, but, as you already mentioned, just being “the most recent penny in the entire bar” could mean very little in terms of the question posed.
They don’t lose the shine just from years, they lose it from being handled.
I’d look for the very rare almost black penny that was issued between our modern pennies conception and 1918. (I forget the exact year it was minted.)
If it is located among those pennies, then I’d make an offer for the bar’s contents to the owner.
Hm. Could it be removed from the resin without damage?
The answer is 2022, without looking at any of the pennies. Let’s say that you examine all of the pennies and find that there is one penny with a date of 2021, and no pennies minted later than 2021. There would be no way to know whether the bar was constructed in the year 2021 or the year 2022. Even if the none of the pennies is newer than, say, 1954, we could not determine the age of the bar going by that date, as the bar could have been constructed at any time after 1954. But the OP assumes that we can guess when the bar was constructed. Since there is no way for us to know the age of the bar going by the date of the newest penny, and Hank assumes we are able to guess that date, I conclude that Hank has given us a trick question. :)
Don’t sweat it. I find people treat statistics like a religion. That is, you’ve got the cheerful atheists who say all statistics is bunk, the unlettered laity who rely on rules of thumb to get by, the rigid clergy who know the rules and apply them mechanically with no regard for why these rules are in place, the charlatans who make it big selling false hope to a population desperate for truth, and occasionally the true believer who love statistics, applies them to himself and gets improvements, and doesn’t quite get why no one else has the same luck with it that he does.
Oh, and the heretic around the edges who’s studied the subject painfully little but enough to come up with questions that are either really stupid or dangerous. That’s where I fall in this cosmogeny.
The imperatives of life and death matter little when there are shiny things to distract.
You can’t live your whole life in fear.
Speaking of heresy; all confidence intervals are made up.
What do you suppose the odds of that are? It’s a good point, so let’s quantify it using reasonable considerations, not galactic improbabilities.
In fact, the bar could be an AI-generated image.
What are the odds that some gap exists such that a number of years before the building of the bar are not represented under the epoxy?
Heck, I’ll give you a starter: perhaps the contractor who does these things acquired a million pennies (say, ten) years before the bar was built and has been drawing upon that stash ever since. This is a reasonable assumption GIVEN your condition. It is however NOT a reasonable assumption without it.
In my own work (ALL DAY TODAY), I have found it necessary to ignore one reasonable assumption (that the then-current year pennies may not have been available, or available in quantity, before the bar was built. I ignore this by assuming that the bar took a year to build, and then assuming that by ignoring both of these considerations, I no longer have to worry about anything less than a year.
Yes, we could go into more detail, but then we would have to show the math. Be my guest.
I collected pennies as a kid. I didn’t do it systemically by date and mint like most folks; I got a jar and filled it with more and more. Then I would dump them out onto the carpet and build pyramids. A nxnx10 pyramid made out of stacks of ten pennies each you can start at the bottom and build up. I know I managed a 6×6 pyramid that way ($9.10), I don’t know if I ever made it to 7×7 ($13.00). Then there was the nxnxn pyramid, which has one penny in the top layer, eight in the next down, and twenty-seven in the third. Though the curve as those reached the top was neat they were always a little frustrating as you had to lay down the bottom layer first, and once you hit the top you were done and couldn’t add any more layers on.
The Swinging Door in Tustin, California.
Every question is a trick question if you refuse to believe in yourself.
I did see one wheat penny in the mix. Couldn’t tell you what year as I saw the wheat side and not the date side.
When they were tearing up my street this past summer there was a random guy out waving a metal detector around. He found a (if memory serves) 1870 Indian head penny.
My street this past summer:
Some baseline assumptions which do not need explaining:
And remember that the questions asks “Judging solely by the dates these pennies were minted, what year was this bar constructed?” An objection such as a gap in the penny record for some years, or that a one-armed man swiped half of them according to a diabolical formula known only to him is special pleading, and must be justified.
Most are, in business, unfortunately, and I would say most+most^2 in government. But it doesn’t have to be that way — provided we are not competing with divine knowledge or time travelling consultants.
A CI is not a feeling, despite the fact that confidence is. A CI is a quantization of expectation given other reasonable assumptions. There are different religions about how CIs, frequency, and probabilities line up.
Heretic.
Starting at one end, we just check one penny. Probably we did not see the date of construction.
Then at the other end, we check 9,999 pennies. Probably got it.
We ignore 0 and 10,000 for now, as we are obviously using “what is the date of the latest penny under the bar” as a proxy for the date the bar was built — based on reasonable assumptions which require no explanation.
If we are Bayesians (and let’s face it — our species is Bayesian except when we educate ourselves out of it), we have a guess in mind anyway, and then we adjust it based on new information. So realistically, we would look around at the bar and then use the single-penny information as a no earlier than date to increase our confidence in our guess. How much would you wager on being right?
Likewise, if we check 9,999 pennies, we will adjust our no-later-than date as required, and probably nail it. In this case, having thousands of times more information than our original guess, we defer utterly to the story told by 9,999 pennies.
But we are restricted by the question not to guess, so our estimate MUST be based ONLY on the evidence presented by the dates on the pennies — no not even the shininess (shame, Hank, your third question cannot be reconciled with the first and main question). Say ten Hail Markovs.
If there’s even one 2022, then we know the age of the bar, don’t we? :)
I keep meaning to write a post about Markov chains, but they’re kind of useless if you don’t know how to multiply matrices, and nobody’s going to care about multiplying matrices unless they need to work out Markov chains. I mean, there are other things you multiply matrices for but few are as straight forward and practical as Markov chains. It’d end up as a long, math filled post where one and all are thoroughly sick of the subject by the time I’m done talking about it, or split into a pair of posts where nobody reads the first one and consequently the second is useless to them.
Given the reasonable assumptions, this problem is another way of asking “what are the odds of drawing a coin of a given age from a supply of 10,000 randomly drawn from circulation?” and then using it for a special case — the newest age.
So what is the distribution like? That’s hard. It’s easier to set up some curves to proxy as the distribution, just to get warmed up:
In all cases, assuming 100 years of coins, and for a 90% CI:
I’ve seen the results of one group trying to do something with the age of coins, and that looked like a hybrid between a 1st and 2nd order function, but really differentiated (so to speak) into zones.
My 20x for 90% is probably NOT a good estimate. Refining (Bayesian me) my answer based on this information, I will say that a hundred will probably do, with a CI of 90%.
Which is EXACTLY what @arizonapatriot said, assuming we may ignore resample.
More to follow.
Nope. Could have been built this morning. See, he’s got us all in a box!
Well, I’m pretty sure it wasn’t built after the photo was taken, so there’s at least that.
Ah, but you might be a Boltzmann brain, only believing that you saw it yesterday.
Ready for a ride? Given what I know right now and without reference to any textbooks or the internet or anything, this is how I’d go about inventing a solution to the problem.
I took a spare change example of 32 pennies last night and constructed a histogram of them. Here it is drawn out:
The whale drawing is my intuition as to how a sample of circulated pennies behaves. From right to left you have zero pennies from the future, a large number of pennies minted in recent years, with a tapering down of frequency from recent years to a long tail of older pennies which can be dismissed as flukes.
With this specific histogram I’m using shininess to determine where to cut off outliers; anything that’s dull is going to be in the tail end and not have much to say about where my whale’s nose is. Calculating the average date of the 19 pennies on the right it’s between 2011 and 2012. That doesn’t tell me much. The median year is 2019, which is where I’ve placed the whale spout. Again I’m working off of intuition here, which is that you could expect a convex slope from the median down to the present year, or rather the year of the bar’s construction in the originally posted question. Even if it don’t look particularly convex from the histogram. (This goes along with Flicker’s intuition that you won’t see many pennies of the current year because they haven’t had time to circulate for long. Guamanian that he is it’s probably a lot more severe that far out.)
To solve this problem I’m mostly interested in the curve to the right of the median. Therefore I need to know what that median is, and what I should expect that curve to look like. To know what the median is I need to know my fluke cutoff, which I’m using the shininess of the pennies as a proxy for. (And if I find 9th District Neighbor’s still shiny wheat pennies I’ll assume they’re shiny because they’ve got a good reason to be.) So, take a sample of shiny pennies from the bar, calculate a median based off of that, and from that median and what I know of the whale snout curve figure out where it hits ‘zero’. That’ll be the year the bar was made.
To get a really good idea of how the whale snout curve for circulating pennies looks like I’d take $10, cash it in for pennies, and construct a really large whale histogram, which I’d then use to try and characterize that curve. Which is counterproductive if I’m curious just about that one bar, but then I’d have the method to look at any bar, shower, or ladies’ bathroom made with this technique and get an answer with a small sample.
Precisely how small? I don’t know; I’ll get back to you when I’ve characterized the whale snout curve.
Just to be clear, the flukes are a consolidation, right? Not binned the same as the rest of the graph?