The Bare Bones of Bayes’ Theorem

February 10, 2012 by · Leave a Comment

Thomas Bayes died over 200 years ago, but his legacy is still with us and provides some very useful insights into probability. What is his legacy? It is a probability formula that tells us how to update probabilities in light of new information.

Suppose, for instance, you learn that Fred is a physical fitness fanatic who works out in the weight room at 7:00pm every Monday, Wednesday, and Friday. Let’s say he’s been doing this for five years straight, without missing a workout. Suppose today is Monday. It’s morning. What’s the probability that Fred will be working out in the weight room tonight? Pretty high, you say, even close to 1.

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The Most Interesting Number

February 3, 2012 by · Leave a Comment

Are all numbers interesting?

Yes.

And we can prove it using a “proof by contradiction.”

  1. Assume there are positive numbers that are not interesting.
  2. Therefore, there is a smallest number that is not interesting.
  3. Hey! That’s interesting!
  4. Therefore, the assumption in (2) is violated. We have a contradiction and have proved that all positive numbers are interesting.

This proof is a bit whimsical. Here’s a similar, more serious question called Berry’s Paradox. [1] Define a positive number X as follows.

“The smallest positive integer not definable in under eleven words.”

This sentence has ten words. And it defines X. Therefore the sentence “The smallest positive integer not definable in under eleven words” defines X in 10 words. Think about it. The statement contradicts itself. We have defined the smallest positive integer that can’t be described in ten words using ten words. This is a contradiction. A paradox.

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Filed Under: Numbers ·

Savvy Advice for Unsavvy Gamblers

January 16, 2012 by · Leave a Comment

Risk is an unavoidable feature of life, so in a sense all of us are all the time gambling. For instance, we are placing a bet when we invest money in a new business venture. Many business ventures don’t succeed. Why then do we do it? Most of the time, it’s because we think we’ve got some edge that will increase our chances of success.

But do we really have an edge? Maybe your life’s dream has been to open a barbecue restaurant. You’ve been cooking barbecue for your family and friends for years, and they tell you it’s the best they’ve had. Your brisket has won first prize at the state barbecue cook-off two years running. So you decide to take the plunge and open a barbecue restaurant.

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The Different Number Scales

January 8, 2012 by · Leave a Comment

When you encounter a number, what sort of numerical information is it giving you? Is the number functioning merely as a label? Or is it counting how many? Does it denote a magnitude or merely indicate an ordering relation? All such questions address the issue of scale. When you see a number, it belongs to one of five scales:

Nominal scale. Here the numbers are merely serving as a label to distinguish different items. Think of social security numbers. They simply distinguish people in the United States, nothing more. The fact that one social security number is bigger than another tells us nothing about people’s respective ages, heights, or anything else about the people who possess them. Social security numbers belong to a nominal scale.

Ordinal scale. Here the numbers indicate an order but provide no magnitude information. For instance, imagine that prisoners in a prison get their numbers assigned by the order in which they entered the prison. The first prisoner gets number 1, the second number 2, and so on. The prisoners’ numbers thus tells us whether one prisoner entered prison before or after another, but tell us nothing about when or how much longer one prisoner entered prison before another. These prison numbers belong to an ordinal scale.

Interval scale. Here the numbers provide information of relative magnitudes. Consider street addresses in a city that is laid out as a grid, such as Chicago. A difference of 800 indicates one mile. Thus Western Avenue, which is 2400 west, is two miles from Pulaski Avenue, which is 4000 west (both these streets run north-south). But allowing 800 to indicate one mile is arbitrary, as is assigning Western Avenue to 2400 west. Instead of 800, city planners could have let 500 indicate a mile; and they could have let Western Avenue be 0 west or even 600 east. The functionality of Chicago street numbers is unchanged by both the addition and the multiplication of a constant, which leave interval information unchanged. These street numbers therefore belong to an interval scale.

Ratio scale. Here the numbers provide information about relative magnitudes, but the zero point is fixed. Consider distance from a given point. One can measure the distance in meters or feet or miles; all that’s required to make the conversion of units is multiplication by a constant. But the distance from a point to itself is always zero and doesn’t depend on the units of measurement. Distance information in this case is unchanged by the multiplication of a constant, which is the ratio of units. Distance therefore belongs to a ratio scale.

Absolute scale. Here the numbers can be only what they are. Such numbers are used to count the exact number of objects in some collection. How many marbles are in that jar? The answer admits only one number, that is, the exact number of marbles in the jar. When a number gives an exact count, it admits no variation. It is what it is and can be no other. In that case, the number belongs to an absolute scale.

REFERENCE: Warren S. Torgerson, Theory and Methods of Scaling. New York: Wiley, 1958.

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The Bare Bones of Probability

December 20, 2011 by · Leave a Comment


How likely is it that an event will happen? Probabilities attempt to answer this question by assigning numbers to the likelihood of events. A probability is always a number between 0 and 1. The closer to 0 the probability, the less likely the event; the closer to 1, the more likely the event. An event with zero probability is thus regarded as impossible, an event with probability equal to one thus is regarded as certain.

When an event occurs, it is one of a range of possibilities. Consider a die with six faces. The range of possibilities here is any of the die’s six faces, each of which has probability 1/6 and is thus as likely to appear as any other. This range of possibilities can be represented by the set {1, 2, 3, 4, 5, 6}. An event can be thought of as any subset of this set.

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The Numbers We Live By

December 7, 2011 by · Leave a Comment

Numbers are an essential part of life, so we better learn to live with them. It’s often said that the language of nature is mathematics, so if you’re going to be a scientist, you’re going to need math. True enough. But non-scientists need numbers too. Keeping a bank account, getting a mortgage, and providing car insurance all require keeping track of numbers. Nor can you get by without probability and statistics. Probability and statistics tell us how to manage risk.

All of life involves risks, but some risks are a lot worse than others. For every 100 million miles driven on American roads by cars, one driver dies. But for every 3.3 million miles driven on American roads by motorcycles, one rider dies. Your chances of dying are thus 30 times more likely on a motorcycle than in a car. Maybe that’s a risk you didn’t know. Maybe you don’t care. But your spouse, who’s taking out life insurance on you, just may.

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Filed Under: Numbers, Statistics ·