Carl, have you ever seen this?

First, she won $5.4 million, then a decade later, she won $2million, then two years later $3million and in the summer of 2010, she hit a $10million jackpot.

The odds of this has been calculated at one in eighteen septillion and luck like this could only come once every quadrillion years.

Harper’s reporter Nathanial Rich recently wrote an article about Ms Ginther, which calls the the validity of her ‘luck’ into question.

First, he points out, Ms Ginther is a former math professor with a PhD from Stanford University specialising in statistics.

A professor at the Institute for the Study of Gambling & Commercial Gaming at the University of Nevada, Reno, told Mr Rich: ‘When something this unlikely happens in a casino, you arrest ‘em first and ask questions later.’

Math pays.

The number 6174 is a really mysterious number. At first glance, it might not seem so obvious. But as we are about to see, anyone who can subtract can uncover the mystery that makes 6174 so special.

Kaprekar’s operation

In 1949 the mathematician D. R. Kaprekar from Devlali, India, devised a process now known as Kaprekar’s operation. First choose a four digit number where the digits are not all the same (that is not 1111, 2222,…). Then rearrange the digits to get the largest and smallest numbers these digits can make. Finally, subtract the smallest number from the largest to get a new number, and carry on repeating the operation for each new number.

It is a simple operation, but Kaprekar discovered it led to a surprising result. Let’s try it out, starting with the number 2005, the digits of last year. The maximum number we can make with these digits is 5200, and the minimum is 0025 or 25 (if one or more of the digits is zero, embed these in the left hand side of the minimum number). The subtractions are:

5200 – 0025 = 5175
7551 – 1557 = 5994
9954 – 4599 = 5355
5553 – 3555 = 1998
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174

Fascinating.

Having determined to write the Principia ten years earlier in 1900, Russell was at first stymied by his discovery of the famous paradox that now bears his name: Consider the set of all those sets that don’t contain themselves. Call this set R. Does R contain itself? If so, it belongs to the set of all sets that don’t contain themselves, and therefore does not contain itself. Does it fail to contain itself? If so, it fails to belong to the set of all sets that don’t contain themselves, and therefore contains itself. Either way, something’s screwy.

Carl?

Maurice Allais, a Nobel prize winning economist, died earlier this month. In this post, I’m going to focus on one of his many intellectual contributions, as it profoundly influenced modern psychology. It’s known as the Allais Paradox, and it was first outlined in a 1953 Econometrica article. Here’s an example of the paradox:

Suppose somebody offered you a choice between two different vacations. Vacation number one gives you a 50 percent chance of winning a three-week tour of England, France and Italy. Vacation number two offers you a one-week tour of England for sure.

Not surprisingly, the vast majority of people (typically over 80 percent) prefer the one-week tour of England. We almost always choose certainty over risk, and are willing to trade two weeks of vacation for the guarantee of a one-week vacation. A sure thing just seems better than a gamble that might leave us with nothing. But how about this wager:

Vacation number one offers you a 5 percent chance of winning a three week tour of England, France and Italy. Vacation number two gives you a 10 percent chance of winning a one week tour of England.

In this case, most people choose the three-week trip. We figure both vacations are unlikely to happen, so we might as well go for broke on the grand European tour. (People act the same way with lotteries: we typically buy the ticket for the biggest possible prize, regardless of the odds.)

Integrals are often described as finding the “area under the curve”. This description is misleading, like saying multiplication is for finding “the area of a rectangle”. Finding area is a useful property, but not the purpose. Integrals help us combine numbers when multiplication can’t.

A logarithmic scale bar. Picking a random x position on this number line, roughly 30% of the time the first digit of the number will be 1 (the widest band of each power of ten).

Benford’s law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises whenever a set of values has logarithms that are distributed uniformly, as is approximately the case with many measurements of real-world values.

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions change.

Anyone with a math background care to comment?