Fermi-type problem: coins and probability

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skullturf
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Fermi-type problem: coins and probability

Postby skullturf » Thu Dec 02, 2010 3:27 pm UTC

I live in Delaware. I always have some coins in my possession -- the amount and composition fluctuates as I go shopping, do laundry, use vending machines, etc.

About how often do I own a coin that was once owned by Hillary Clinton?

(Obviously this isn't a pure math problem, but I am interested to see order-of-magnitude estimates using plausible assumptions.)

Edit: I figure 100 is probably an okay round-number estimate of the number of new (to me) coins that come into my possession in a month. (I currently have 150 pennies in rolls in my house, all of which I collected in the last three months.)

Some of you may have heard about the "Where's George?" project that attempts to track dollar bills. We might try to use their data to make plausible guesses about the travel patterns of individual bills or coins.

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Yakk
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Re: Fermi-type problem: coins and probability

Postby Yakk » Fri Dec 03, 2010 5:05 pm UTC

Coins have a half-life. Call it 30 years (every 30 years, half of the coins in circulation will have been lost or recycled).

That's about a 0.998 decay rate per month.

We'll assume Hillary also has a 100 coin/month flow rate, and had this rate for 50 years.

sum(i = 0 to infinity) (100 * 0.998^i) = 100 * (1-.998^600)/(1-.998) =~ 34958.4 coins that have been previously owned by HC in circulation.

Call it 35000.

The US economy is about 10 trillion dollars per year. Suppose 1% of this is a cash economy, of which 5% is coins. (does this line up with 100 new coins/month?)

5 billion $ in coins. At 100 coins/month, and ~200 million adults, that is a coin value*velocity factor of 0.25. Which doesn't seem that far off.

We'll give coins a velocity of 2.5 (annual), so the average value of a coin is 10 cents. Thus there are 50 billion US coins in circulation.

35 thousand / 50 billion = 0.7 E-6

100 coins/month = 1.2 E4 per year.

Average number of HC coins/year = 0.84E-2.

Average of 120 years between HC owned coins.

...

Issue: with ~300 billion US citizens, 50 billion coins seems low. (even if you ignore coins that are "mostly out of circulation", like ones in your cushions)

Having more coins makes the situation worse, however. So the answer is ... you won't and never will own such a coin.

...

Improvements: We can model the average coin size better, based off the assumption that coins are mainly used for change on relatively random values. We could determine coins in circulation via an approximation of coin velocity and size, checking against economy size for sanity. This improves the # of coin estimate, almost certainly upwards, and reduces likelyhood of you having a coin.

...

Assumption flaw: 100 coins/month might be way off. An increase in this rate has quadratic impact on your chances (unless it also impacts the population of coins in existence). The half-life of 30 years, but that has a pretty small impact (a factor of 2, basically).

...

A different question: have you owned a coin that was co-owned with a coin once owned by HC? (ie that "touched" in a coin-purse/pocket) How often?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

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Blatm
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Re: Fermi-type problem: coins and probability

Postby Blatm » Fri Dec 03, 2010 5:24 pm UTC

Yakk wrote:35 thousand / 50 billion = 0.7 E-6

100 coins/month = 1.2 E4 per year.

Average number of HC coins/year = 0.84E-2.

Average of 120 years between HC owned coins.


That should be 1.2 E3 per year, not 1.2 E4. That changes the answer from 120 years/coin to 1200 years/coin.

I think 50 billion coins for 300 million people (150 coins/person) seems pretty good. It's hard to tell, because I suspect it varies pretty wildly from person to person. I suspect it would be a bit higher than 150/person because people with fewer than 150 coins probably don't have that few, but it's fairly easy to have 5 times 150. I'd be surprised if 150 weren't good to half an order of magnitude though. Otherwise, 100 coins/month might be a bit low. Is it low by the same factor 150 coins/person is low? That factor would cancel.

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Velifer
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Re: Fermi-type problem: coins and probability

Postby Velifer » Fri Dec 03, 2010 8:47 pm UTC

Your plan isn't going to work. You won't be able to get enough DNA to grow a clone and replace her.

Could this be better modeled as a counting problem? There are only so many pockets for those coins to be in. Their movements are essentially a Markov web.

You just need to figure out the path with the largest probability... Or influence this by mowing her lawn.
Time flies like an arrow, fruit flies have nothing to lose but their chains -Marx


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