{25,25,25,10,10,1,1,1,1}In the cold weather months, I'll usually have change for any fractional part of a dollar, since my coats have large pockets in which I carry a lot of coins. In that case, I'm just like an athlete who wears training weights. I've never examined the distribution of coins in my coin stash, but I'm sure that it would follow some statistical rule. Two mathematicians have posted a paper on arXiv to answer the question "What set of coins is most likely to be the contents of your wallet?" They are Lara Pudwell of the Department of Mathematics and Computer Science, Valparaiso University (Valparaiso, Indiana) and Eric Rowland of the Laboratoire de combinatoire et d'informatique mathématique, Université du Québec à Montréal (Montreal, Canada).[2-3] Their calculation was a little more involved than my simple program, since they looked at all ways that fractional parts of a dollar that can be returned as change from a purchase, not just the ninety-nine cent case. Their assumptions were that there's a uniform distribution of prices between 0 and 99 cents and cashiers return change as the smallest number of coins. As in my example, half-dollar coins were disallowed. exercise for the reader, which is what my professors and textbook authors always said. When I was a child, I could purchase "penny candy." Now, a penny wouldn't even buy you the right to sniff a chocolate bar, since inflation has rendered pennies practically worthless. In 1982, when the copper value of a US penny was nearly a penny, the US Mint started making pennies which are 97.5% zinc with a 2.5% copper plating. Still, the manufacturing and distribution cost of a US penny is about two cents.[4] People don't usually carry pennies, and they drop them into the popular give-a-penny/take-a-penny cups at cashiers. So, these shoppers are "pennyless," as distinct from "penniless," and they require a different analysis. The following table shows the ten most likely wallet states for the pennyless purchaser. Note that even the most probable state, that of an empty wallet, has just a five percent likelihood.

{10,10,5,25,25,10,10,1,1,1,1}

{10,10,5,10,10,5,25,10,10,1,1,1,1}

{10,10,5,10,10,5,10,10,5,10,10,1,1,1,1}

{5,5,10,5,10,10,5,10,10,5,10,10,1,1,1,1}

{5,5,5,5,5,10,10,5,10,10,5,10,10,1,1,1,1}

{5,5,5,5,5,5,5,10,5,10,10,5,10,10,1,1,1,1}

{5,5,5,5,5,5,5,5,5,5,10,10,5,10,10,1,1,1,1}

{5,5,5,5,5,5,5,5,5,5,5,5,10,5,10,10,1,1,1,1}

{5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,10,10,1,1,1,1}

{5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,10,1,1,1,1}

{5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,1,1,1,1}

...etc., for 31 total combinations.

State | Probability | State | Probability | |

{empty} | .05000 | {10, 5, 5} | .02731 | |

{5} | .05000 | {25, 25, 10, 5} | .02625 | |

{10, 5} | .03916 | {5, 5} | .02536 | |

{25, 10, 5} | .03093 | {10} | .02463 | |

{25, 5} | .02847 | {25, 10, 5, 5} | .02417 |

• There should be ten coins.My cold weather pockets contain enough in coins that I always have exact change. I suspect that keeping about two dollars in coins, perhaps weighted on the penny side (in more ways than one), is enough to ensure that fact. Pudwell and Rowland also calculate the approximate distribution of circulating coins as 10.5% quarters, 11.5% dimes, 9.1% nickels, and 68.9% pennies. The distribution of U.S. minted coins in 2012 is actually 6.1% quarters, 18.1% dimes, 11.0% nickels, and 64.8% pennies.[2] Not bad for an

• The probability that there is at least one nickel is 0.58085.

• The probability that there is at least one penny is 0.95975.

• The probability of there being only pennies, if you have any coins at all, is 0.08430.

• The probability of having an empty wallet is .01000.

• The probability of having the exact change is only 0.00831.

• The expected number of quarters is 1.06.

• The expected number of dimes is 1.15.

• The expected number of nickels is 0.91.

• The expected number of pennies is 6.92.

- H. Bethe, "Termaufspaltung in Kristallen," Annalen der Physik, vol. 395, no. 2 (1929), pp. 133-208. Translation appears in Splitting of Terms in Crystals, chap. 1 of Selected Works of Hans A. Bethe: With Commentary, (World Scientific, Jan 1, 1997), 605 pages.
- Lara Pudwell and Eric Rowland, "What's in your wallet?!" arXiv Preprint Server, June 9, 2013.
- Evelyn Lamb, "Mathematicians Predict What's in Your Wallet," Scientific American Blogs, June 20, 2013.
- Michael Zielinski, "Cost to Make Penny and Nickel Declines But Still Double Face Value," Coin Update, December 10, 2012.
- Bill Gates, The World's Billionaires (#2), Forbes, March, 2013.