Tag Archives: mathematics

An Analysis of Supermarket Checkout Times

An analysis of supermarket checkout times has shown that express lanes (for people with fewer than 5 items, say) are not always the most efficient checkout route for time-sensitive shoppers.

Dan Meyer, a high school maths teacher, has done the hard work (providing his data and analysis) and came to the following conclusion:

[Express lanes] attract more people holding fewer total items, but as the data shows […], when you add one person to the line, you’re adding 48 extra seconds to the line length (that’s “tender time” added to “other time”) without even considering the items in her cart. Meanwhile, an extra item only costs you an extra 2.8 seconds. Therefore, you’d rather add 17 more items to the line than one extra person!

via Kottke

The Story of Big Numbers

Physicist Albert Bartlett is quoted as saying that “the greatest shortcoming of the human race is our inability to understand the exponential function”.

Starting with a thought experiment in which two competitors are challenged to come up with the bigger finite number, Scott Aaronson has written an accessible and fact-filled essay about large numbers, touching on topics such as AI, NP problems, the computational power of our brains, and much more besides.

Place value, exponentials, stacked exponentials: each can express boundlessly big numbers, and in this sense they’re all equivalent. But the notational systems differ dramatically in the numbers they can express concisely. […] It takes the same amount of time to write 9999, 9999, and 9999—yet the first number is quotidian, the second astronomical, and the third hyper-mega astronomical. The key to the biggest number contest is not swift penmanship, but rather a potent paradigm for concisely capturing the gargantuan.

Such paradigms are historical rarities. We find a flurry in antiquity, another flurry in the twentieth century, and nothing much in between. But when a new way to express big numbers concisely does emerge, it’s often a by-product of a major scientific revolution: systematized mathematics, formal logic, computer science. Revolutions this momentous, as any Kuhnian could tell you, only happen under the right social conditions. Thus is the story of big numbers a story of human progress.

This essay inspired the 2007 Big Number Duel at MIT.

Learn Statistics, Damn You!

Thanks to my moderate knowledge of statistics, I know that I have a lot more to learn in the field and should never make assumptions about data or analyses (even my own).

Because of this I share a grievance with Zed Shaw who says that “programmers need to learn statistics or I will kill them all”. Required reading and advice not just for programmers, but for everyone who looks at data, creates models, or even reads a newspaper.

I have a major pet peeve that I need to confess. I go insane when I hear programmers talking about statistics like they know shit when its clearly obvious they do not. I’ve been studying it for years and years and still don’t think I know anything. This article is my call for all programmers to finally learn enough about statistics to at least know they don’t know shit. I have no idea why, but their confidence in their lacking knowledge is only surpassed by their lack of confidence in their personal appearance.

My recommendation? Read this article to realise that you know nothing, and then pick up a copy of John Allen Paulos’ Innumeracy and Darrell Huff’s How to Lie with Statistics in order to realise that you know even less than you thought (but a hell of a lot more than the average person).

Being Rational About Risk

Leonard Mlodinow—physicist at Caltech and author of The Drunkard’s Walk, a highly-praised book looking at randomness and our inability to take it into account—has an interview in The New York Times about understanding risk. Some choice quotes:

I find that predicting the course of our lives is like predicting the weather. You might be able to predict your future in the short term, but the longer you look ahead, the less likely you are to be correct.

I don’t think complex situations like [the current financial crisis] can be predicted. There are too many uncontrollable or unmeasurable factors. Afterwards, of course, it will appear that some people had gotten it just right: since there are many people making many predictions, no doubt some of them will get it right, if only by chance. But that doesn’t mean that, if not for some unforeseen random turn, things wouldn’t have gone the other way. […]

In some sense this idea is encapsulated in the cliché that “hindsight is always 20/20,” but people often behave as if the adage weren’t true. In government, for example, a “should-have-known-it” blame game is played after every tragedy.

As someone who has taken risks in life I find it a comfort to know that even a coin weighted toward failure will sometimes land on success. Or, as I.B.M. pioneer Thomas Watson said, “If you want to succeed, double your failure rate.”

I haven’t had a chance to watch it, but in May 2008 Mlodinow spoke for the [email protected] series.

The Birthday Problem

I’ve heard of this ‘problem’ numerous times before, as I’m sure many others have too. Nonetheless, everytime I do hear it, it fascinates me.

The birthday problem (or paradox, as it’s often referred), looks at the probability of two or more people from a randomly chosen set of people sharing a birthday.

In a group of at least 23 randomly chosen people, there is more than 50% probability that some pair of them will both have been born on the same day. For 57 or more people, the probability is more than 99%, and it reaches 100% when the number of people reaches 367[…]. The mathematics behind this problem leads to a well-known cryptographic attack called the birthday attack.