# Difference between revisions of "Chance News 62"

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<blockquote>This article reports on the work of economist Patrick DeJarnette, who developed some unusual results in probabilistic integer arithmetic. We can combine pennies, nickels, dimes and quarters to pay for any item costing below a dollar (freebies included, so (0.99 cents) is the relevant range). With the assumption that each of these 100 prices is equally likely to occur and that a purchaser uses the fewest possible coins, then on average each purchase requires 4.70 coins. DeJarnette asked if some other set of four coins, perhaps including one worth 8 or 61 or 37 cents would yield a lower average. He found that the optimal result uses an average of 4.10 coins per purchase, and is achieved by the sets | <blockquote>This article reports on the work of economist Patrick DeJarnette, who developed some unusual results in probabilistic integer arithmetic. We can combine pennies, nickels, dimes and quarters to pay for any item costing below a dollar (freebies included, so (0.99 cents) is the relevant range). With the assumption that each of these 100 prices is equally likely to occur and that a purchaser uses the fewest possible coins, then on average each purchase requires 4.70 coins. DeJarnette asked if some other set of four coins, perhaps including one worth 8 or 61 or 37 cents would yield a lower average. He found that the optimal result uses an average of 4.10 coins per purchase, and is achieved by the sets | ||

(1, 3, 11, 37,) or (1, 3, 11, 38). </blockquote> | (1, 3, 11, 37,) or (1, 3, 11, 38). </blockquote> | ||

+ | Norton suggests that "there is a lot of play in this curious study." This is verified by looking at the 116 comment at the end of this Blog. | ||

## Revision as of 01:58, 13 March 2010

## Contents

## Quotations

"It is a very sad thing that nowadays there is so little useless information."

--Oscar Wilde

Quoted in the Economist article on data cited below.

"Statisticians are engaged in an exhausting but exhilarating struggle with the biggest challenge that philosophy makes to science: how do we translate information into knowledge? ...

"If you think that statistics has nothing to say about what you do or how you could do it better, then you are either wrong or in need of a more interesting job."

--Stephen Senn,

Dicing With Death

Suggested by Paul Alper

## Forsooth

## Too much data?

All too much: Monstrous amounts of data

The Economist, 25 February 2010

A special report on managing data. To be continued...

## Media highlights

The College Mathematics Journal has a column called Media Highlights. Norton Starr is one of the editors and his contributions are usually of interest to Chance News readers. In the March issue Noton has two such items.

For Decades, Puzzling People With Mathematics

The New York Times
Stevin D Levitt

March 11, 2010

Norton writes.

This article gives an inspiring portrait of Martin Gardner, the

premier exponent of recreational mathematics for over 50 years.

It invites readers to see mathematics vastly richer and more interesting

than they may recall from their classrooms exercise. Norton says more

about Martin and ends with a quote from Rohnald Graham:

Martin has turned thousands of children into mathematicians,

and thousand of mathematicians into children.

Do We Need a 37-Cent Coin?

The New York Times
John Tierney

October 19,2009

Here Norton Writes:

This article reports on the work of economist Patrick DeJarnette, who developed some unusual results in probabilistic integer arithmetic. We can combine pennies, nickels, dimes and quarters to pay for any item costing below a dollar (freebies included, so (0.99 cents) is the relevant range). With the assumption that each of these 100 prices is equally likely to occur and that a purchaser uses the fewest possible coins, then on average each purchase requires 4.70 coins. DeJarnette asked if some other set of four coins, perhaps including one worth 8 or 61 or 37 cents would yield a lower average. He found that the optimal result uses an average of 4.10 coins per purchase, and is achieved by the sets (1, 3, 11, 37,) or (1, 3, 11, 38).

Norton suggests that "there is a lot of play in this curious study." This is verified by looking at the 116 comment at the end of this Blog.

Submitted by Laurie Snell

## Back of the envelope calculations about Toyota

Toyotas Are Safe (Enough). Robert Wright, The New York Times Blog, March 9, 2010.

How worried are you about driving a Toyota? Robert Wright is not that worried.

My back-of-the-envelope calculations (explained in a footnote below) suggest that if you drive one of the Toyotas recalled for acceleration problems and don’t bother to comply with the recall, your chances of being involved in a fatal accident over the next two years because of the unfixed problem are a bit worse than one in a million — 2.8 in a million, to be more exact. Meanwhile, your chances of being killed in a car accident during the next two years just by virtue of being an American are one in 5,244. So driving one of these suspect Toyotas raises your chances of dying in a car crash over the next two years from .01907 percent (that’s 19 one-thousandths of 1 percent, when rounded off) to .01935 percent (also 19 one-thousandths of one percent).

Of course, the type of risk involved is part of the problem.

But lots of Americans seem to disagree with me. Why? I think one reason is that not all deaths are created equal. A fatal brake failure is scary, but not as scary as your car seizing control of itself and taking you on a harrowing death ride. It’s almost as if the car is a living, malicious being.

Robert Wright includes an appendix with all of the computations and assumptions that went into these numbers.

Submitted by Steve Simon

### Questions

1. People seem to make a distinctions between risks that they place upon themselves (e.g., talking on a cell phone while driving) and risks that are imposed upon them by an outsider (e.g., accidents caused by faulty manufacturing). Is this fair?

2. Contrast the absolute change in risk (.01935-.01907=.00028) with the relative change in risk (.01935/.01907=1.0147). Which way seems to better reflect the change in risk?

3. Examine the assumptions that Robert Wright uses. Do these seem reasonable?