Tuesday, 5 August 2008
Thursday, 31 July 2008
Why are planets called planets?
Ancient astronomers knew that the stars move across the sky keeping together in the same fixed patterns. We call these patterns constellations. In the course of a twenty-four hour period, the constellations return to approximately their original place.
These astronomers were also aware that there were some stars that did not stay in fixed positions relative to the other stars. Instead, from night to night, they seemed to wander about the sky, following peculiar loop-the-loop paths of their own, while the rest of the stars formed a fixed background to their dancing.
The word for these wandering stars that we use now came to us from the Ancient Greeks, who used their own word for wanderer,
These astronomers were also aware that there were some stars that did not stay in fixed positions relative to the other stars. Instead, from night to night, they seemed to wander about the sky, following peculiar loop-the-loop paths of their own, while the rest of the stars formed a fixed background to their dancing.
The word for these wandering stars that we use now came to us from the Ancient Greeks, who used their own word for wanderer,
(planetes)
or planet.
And that's why planets are called planets.
Monday, 28 July 2008
Mark Borkowski's Fame Formula
A public relations agent claims to have come up with a formula which "illustrates" the decline in fame of a brand or celebrity (he treats these two as synonymous) over time.
Mark Borkowski is a regular Guardian columnist and head of Borkowski PR. By happy chance the article in the Guardian where he introduces his formula coincides with the publication of his book today, where he discusses it, and celebrity in general in more detail. I hope the £16.99 price tag is worth it as it's difficult to make sense of his mathematics just from reading the article.
Mr. Borkowski wants us to know that (a) if you have a brand or are a celebrity your fame will decline dramatically over time, and (b) a good publicist can help prevent this and keep you in the public eye by judicious planting of stories and attention-grabbing vignettes. The trouble is that most celebrities and brand managers know this already and wouldn't see the need to purchase Mr Borkowski's services, so it's a good thing Mr Borkowski has developed a quantitative means of assessing fame which will give him the edge over his competitors. This is his fabulous formula:
Where F is the level of fame, T is time measured in three monthly intervals, B is a baseline of fame calculated from the average level of fame before the peak, and P is the "increment" of fame above the baseline, "that establishes the individual firmly at the front of public consciousness".
The formula is said to illustrate "that without intervention in the form of further publicity, fame follows an exponential slide to obscurity." Borkowski goes further than this and states that the slide to obscurity lasts about fifteen months. He shows us this by substituting T=5 into the formula (representing fifteen months) to obtain F = B + 0.04P, showing that the fame-boost received at the height of public attention has been reduced by 96%.
One is tempted to think that the formula is a bit of a gimmick, since publicity and the level of media interest in a brand or celebrity is a hard thing to quantify. Nevertheless, one would expect, given reasonable values for the parameters and the inputs, that the formula would give plausible outputs. However, those who do think this would be reckoning without a shocking ignorance of basic mathematics on the part of Mr Borkowski, and his assumption of the same ignorance on the part of the Great British Guardian reader, for his formula is a load of bollocks. For those who wish for a restoration of mathematical sanity there is the formula that Mr Borkowski should have used at the end of this article. It took about twenty seconds to draw up.
The problem with Mr Borkowski's formula is that it gives nonsensical values across an important part of the range over which it is supposed to apply. Here's an example: What fame value does it give at one month after the initial kick of publicity? At this point T = 1/3. If we plug this into the formula we get the answer F = B + 4.8P which is more than 4.8 times higher than the supposed high point is above the baseline.
There is also a problem in that at T = 0 (the starting point) the formula cannot produce a fame value. As the value of T gets smaller and smaller (approaching zero) the amount of fame gets larger and larger, and approaches infinity. Mr Borkowski thinks that at t = 0 the formula gives an infinite fame value, which he acknowledges is not accurate, but nonetheless thinks appropriate as it puts people in mind of the fame value being "off the radar". This is untrue: the formula does not give a fame value at T = 0 as the operation necessary to calculate it cannot be performed. Mr Borkowski is clearly sacrificing accuracy to appearances here. Further details at the end for those who can stomach it.
Here is a graph showing what Mr. Borkowski wants the formula to do, and the formula he should have used:
And here is graph of the formula he has actually used. Time is in Mr Borkowski's units of three month intervals, notional parameters of a baseline B of 1 and initial kick P of 1 have been used:
Call me a cynic, Mr Borkowski, but isn't your formula part of an avaricious (though mathematically inept) scheme to increase sales of your book by fooling people into thinking you've placed PR on a scientific footing? An A-level maths student could have told you you'd used the wrong function to model the data because it doesn't give sensible answers across the time period, but you went ahead anyway and fooled yourself into thinking that the discontinuities gave it extra cachet, rather than being an indication that you'd got something wrong. The irony is that you would have got away with it if you'd only used the formula given above. How much do you pay your mathematics consultant?
_____________________________________________________________
A More Technical Addendum
Let's look at the formula in more detail. Mr Borkowski does not give us any of the data he obviously used to derive it. He does not tell us any values for the baseline fame B, or the initial increment P given to the celebrity or brand: some sort of example would have been helpful here, because it could have been used to check his claims. But there's another nagging problem that only sad geeky people would be interested in; you see, despite stating that a celebrity's fame follows an exponential slide, this formula is a power function and not an exponential one. An exponential function would look like the formula given above and reproduced here:
where the time variable is in the index or exponent (hence the name). The difference is important because if he had used a function like this Mr Borkowski might have earned himself a little more credibility. The problem is that his original formula is what mathematicians call "undefined" at t = 0, and that's because of the 1/T and 1/T^2 (the ^2 means "to the power of 2" or "squared"). When you substitute zero into the formula wherever T is you end up trying to divide by zero, which is an operation in mathematics that is not allowed.
To see why, remember that dividing one number by another is equivalent to subtracting the second number repeatedly from the first until you reach zero. So 6 divided by 2 gives an answer of 3 because when you subtract 2 repeatedly from 6 you can do it 3 times until you reach zero. Now imagine dividing 6 by zero. It doesn't matter how many times you take 0 away from 6, you won't budge at all from your starting number. Hence you'll never reach zero. The division process is not working and that's why the operation is called undefined. Computers have to be specially programmed to recognise when they're getting into a situation where they'll be asked to divide by zero, or they'd plunge themselves into an endless loop trying to achieve it.
Mark Borkowski is a regular Guardian columnist and head of Borkowski PR. By happy chance the article in the Guardian where he introduces his formula coincides with the publication of his book today, where he discusses it, and celebrity in general in more detail. I hope the £16.99 price tag is worth it as it's difficult to make sense of his mathematics just from reading the article.
Mr. Borkowski wants us to know that (a) if you have a brand or are a celebrity your fame will decline dramatically over time, and (b) a good publicist can help prevent this and keep you in the public eye by judicious planting of stories and attention-grabbing vignettes. The trouble is that most celebrities and brand managers know this already and wouldn't see the need to purchase Mr Borkowski's services, so it's a good thing Mr Borkowski has developed a quantitative means of assessing fame which will give him the edge over his competitors. This is his fabulous formula:
Where F is the level of fame, T is time measured in three monthly intervals, B is a baseline of fame calculated from the average level of fame before the peak, and P is the "increment" of fame above the baseline, "that establishes the individual firmly at the front of public consciousness".
The formula is said to illustrate "that without intervention in the form of further publicity, fame follows an exponential slide to obscurity." Borkowski goes further than this and states that the slide to obscurity lasts about fifteen months. He shows us this by substituting T=5 into the formula (representing fifteen months) to obtain F = B + 0.04P, showing that the fame-boost received at the height of public attention has been reduced by 96%.
One is tempted to think that the formula is a bit of a gimmick, since publicity and the level of media interest in a brand or celebrity is a hard thing to quantify. Nevertheless, one would expect, given reasonable values for the parameters and the inputs, that the formula would give plausible outputs. However, those who do think this would be reckoning without a shocking ignorance of basic mathematics on the part of Mr Borkowski, and his assumption of the same ignorance on the part of the Great British Guardian reader, for his formula is a load of bollocks. For those who wish for a restoration of mathematical sanity there is the formula that Mr Borkowski should have used at the end of this article. It took about twenty seconds to draw up.
The problem with Mr Borkowski's formula is that it gives nonsensical values across an important part of the range over which it is supposed to apply. Here's an example: What fame value does it give at one month after the initial kick of publicity? At this point T = 1/3. If we plug this into the formula we get the answer F = B + 4.8P which is more than 4.8 times higher than the supposed high point is above the baseline.
There is also a problem in that at T = 0 (the starting point) the formula cannot produce a fame value. As the value of T gets smaller and smaller (approaching zero) the amount of fame gets larger and larger, and approaches infinity. Mr Borkowski thinks that at t = 0 the formula gives an infinite fame value, which he acknowledges is not accurate, but nonetheless thinks appropriate as it puts people in mind of the fame value being "off the radar". This is untrue: the formula does not give a fame value at T = 0 as the operation necessary to calculate it cannot be performed. Mr Borkowski is clearly sacrificing accuracy to appearances here. Further details at the end for those who can stomach it.
Here is a graph showing what Mr. Borkowski wants the formula to do, and the formula he should have used:
And here is graph of the formula he has actually used. Time is in Mr Borkowski's units of three month intervals, notional parameters of a baseline B of 1 and initial kick P of 1 have been used:
Call me a cynic, Mr Borkowski, but isn't your formula part of an avaricious (though mathematically inept) scheme to increase sales of your book by fooling people into thinking you've placed PR on a scientific footing? An A-level maths student could have told you you'd used the wrong function to model the data because it doesn't give sensible answers across the time period, but you went ahead anyway and fooled yourself into thinking that the discontinuities gave it extra cachet, rather than being an indication that you'd got something wrong. The irony is that you would have got away with it if you'd only used the formula given above. How much do you pay your mathematics consultant?
_____________________________________________________________
A More Technical Addendum
Let's look at the formula in more detail. Mr Borkowski does not give us any of the data he obviously used to derive it. He does not tell us any values for the baseline fame B, or the initial increment P given to the celebrity or brand: some sort of example would have been helpful here, because it could have been used to check his claims. But there's another nagging problem that only sad geeky people would be interested in; you see, despite stating that a celebrity's fame follows an exponential slide, this formula is a power function and not an exponential one. An exponential function would look like the formula given above and reproduced here:
where the time variable is in the index or exponent (hence the name). The difference is important because if he had used a function like this Mr Borkowski might have earned himself a little more credibility. The problem is that his original formula is what mathematicians call "undefined" at t = 0, and that's because of the 1/T and 1/T^2 (the ^2 means "to the power of 2" or "squared"). When you substitute zero into the formula wherever T is you end up trying to divide by zero, which is an operation in mathematics that is not allowed.
To see why, remember that dividing one number by another is equivalent to subtracting the second number repeatedly from the first until you reach zero. So 6 divided by 2 gives an answer of 3 because when you subtract 2 repeatedly from 6 you can do it 3 times until you reach zero. Now imagine dividing 6 by zero. It doesn't matter how many times you take 0 away from 6, you won't budge at all from your starting number. Hence you'll never reach zero. The division process is not working and that's why the operation is called undefined. Computers have to be specially programmed to recognise when they're getting into a situation where they'll be asked to divide by zero, or they'd plunge themselves into an endless loop trying to achieve it.
Thursday, 24 July 2008
Darwinism
The addition of an -ism suffix to any noun , especially a proper noun, usually puts the idea in people's heads that an ideology is being referred to. Examples are Thatcherism, Marxism, Freudianism, Lysenkoism and Hegelianism. The -ism suffix can also mean an action, process or characteristic behaviour, and examples of these are volcanism and despotism, but for most people, placing -ism on the end of a word is interpreted in its Maureen Lipman sense ("You got an -ology, you're a scientist!") and taken to mean an ideological position.
In fact, this connotation of ideology is so strong that the suffix has, as far back as the seventeenth century, been stripped of its possible antecedents and used as a word in its own right. Type "ism" into the OneLook internet dictionary portal site, and the quick definition returned is "a belief (or system of beliefs) accepted as authoritative by some group or school". The twenty-three ordinary dictionaries that the website turns up as containing "ism" as a word mostly agree on this, and the Webster's Revised Unabridged, 1913 Edition goes as far as to define an "ism" as "a doctrine or theory; especially, a wild or visionary theory".
In science, ideologies and other fixed beliefs are bad things, because they prevent us from testing an idea for its soundness by exposing it to potentially contradictory evidence. Instead, an ideology will usually try and wrap reality around its framework. The scientific method, on the other hand, is all about changing our ideas to fit reality. Failure to do this can lead to disaster, as the Soviet Union found out when it realised that being in the thrall of the bizarre ideas of the fraudulent agronomist Trofim Lysenko for several decades had set back their progress in biotechnology by the same amount of time. A scientifically minded person therefore, has a well-justified fear of ideology.
This fear is implicit in Olivia Judson's recent article in the New York Times, in which she argues that the terms Darwinism and Darwinist should be abolished because they give the false impression that the field was the "brainchild of a single person" many years ago, and that (crucially for an ideology) it has not changed in a significant way since then. Judson recognises, though she does not say explicitly, that the use of the term Darwinism, especially by fundamentalist religious groups, allows the hearer to infer that evolutionary biology is just another ideology, deserving of no more support than any other idea or belief. In this way, such organisations can caricature the theory of evolution, and thus make it easier to attack.
Of course, evolutionary biology is not an ideology: it has changed enormously since the nineteenth century, while still retaining the core ideas that Darwin espoused a hundred and fifty years ago. This is the essence of a field of science, revolutions in understanding occur, but they very rarely change our viewpoint completely. The development of Einstein's general theory of relativity did not wipe out Newton's laws of gravity. Like all good theories, it explained why gravity worked as it did, noted that Newton's equations worked perfectly accurately for most purposes, and predicted that there were occasions where Newton's laws would break down - mainly under high velocities and strong gravities - and provided an explanation of what would happen under these circumstances, time slowing down being one of them. In other words, Einstein's theories superseded those of Newton rather than completely replaced them. The only areas where a complete change of viewpoint took place were where the previous ideas were religious or ideological in nature. Examples of these are the heliocentric view of the solar system, or the development of modern medicine.
In fact, this connotation of ideology is so strong that the suffix has, as far back as the seventeenth century, been stripped of its possible antecedents and used as a word in its own right. Type "ism" into the OneLook internet dictionary portal site, and the quick definition returned is "a belief (or system of beliefs) accepted as authoritative by some group or school". The twenty-three ordinary dictionaries that the website turns up as containing "ism" as a word mostly agree on this, and the Webster's Revised Unabridged, 1913 Edition goes as far as to define an "ism" as "a doctrine or theory; especially, a wild or visionary theory".
In science, ideologies and other fixed beliefs are bad things, because they prevent us from testing an idea for its soundness by exposing it to potentially contradictory evidence. Instead, an ideology will usually try and wrap reality around its framework. The scientific method, on the other hand, is all about changing our ideas to fit reality. Failure to do this can lead to disaster, as the Soviet Union found out when it realised that being in the thrall of the bizarre ideas of the fraudulent agronomist Trofim Lysenko for several decades had set back their progress in biotechnology by the same amount of time. A scientifically minded person therefore, has a well-justified fear of ideology.
This fear is implicit in Olivia Judson's recent article in the New York Times, in which she argues that the terms Darwinism and Darwinist should be abolished because they give the false impression that the field was the "brainchild of a single person" many years ago, and that (crucially for an ideology) it has not changed in a significant way since then. Judson recognises, though she does not say explicitly, that the use of the term Darwinism, especially by fundamentalist religious groups, allows the hearer to infer that evolutionary biology is just another ideology, deserving of no more support than any other idea or belief. In this way, such organisations can caricature the theory of evolution, and thus make it easier to attack.
Of course, evolutionary biology is not an ideology: it has changed enormously since the nineteenth century, while still retaining the core ideas that Darwin espoused a hundred and fifty years ago. This is the essence of a field of science, revolutions in understanding occur, but they very rarely change our viewpoint completely. The development of Einstein's general theory of relativity did not wipe out Newton's laws of gravity. Like all good theories, it explained why gravity worked as it did, noted that Newton's equations worked perfectly accurately for most purposes, and predicted that there were occasions where Newton's laws would break down - mainly under high velocities and strong gravities - and provided an explanation of what would happen under these circumstances, time slowing down being one of them. In other words, Einstein's theories superseded those of Newton rather than completely replaced them. The only areas where a complete change of viewpoint took place were where the previous ideas were religious or ideological in nature. Examples of these are the heliocentric view of the solar system, or the development of modern medicine.
Saturday, 29 December 2007
Voice Risk Analysis Software criticised in national media
Regular readers will remember my writing an article about voice risk analysis software employed by Harrow Council in particular to detect benefits claimants who may be lying. At the time I pointed out that there was no evidence, other than anecdotal, that the product worked as claimed. And that:
"Harrow Council, and soon the Department of Work and Pensions, could well be detecting and prosecuting fraudsters using software whose only claim to be efficacious is that the company that makes it says it is."
Today I find that this is not just my opinion, but also that of Paul Lewis, presenter of Radio 4's Moneybox programme. He popped up on Breakfast on BBC1 this morning to put the arguments against the credibility of these devices much more succinctly than I have done.
In brief, Voice Risk Analysis (VRA) software claims to detect stress in a person's voice, not lying specifically. In day to day use, the software flags up calls from benefit claimants whose voices betray signs of stress and the call can then be followed up with further checks.
The presenters told Lewis that Harrow Council had saved £110,000 by the use of Voice Stress Analysis: Lewis pointed out that the problem was that they had nothing to compare the method to, how did Harrow Council know for instance, that picking people at random and following up their claims would not generate an equivalent saving? Also, simply knowing that Harrow Council was piloting such a scheme might be enough to put off fraudulent claimants (as well as discouraging genuine claimants wary of harassment).
Superbly put. Excellent piece. Also in Lewis's favour was his mention of Niels Bohr in the context of making predictions about the future. Anyone who can mention a famous physicist at 8:45 in the morning on national television gets my vote.
A piece on VRA also appeared later in the day on the 29th December issue of Moneybox. Lewis was presenting and went into a lot more detail.
It turns out that the supplier of the software is Digilog as predicted on this blog back in September. Digilog, it seems, will not tell anyone how it is able to detect stress in voices, nor why they assume that stress and lying are linked. This should start alarm bells ringing in any reasonable person's head. Contrary to what your intuition might tell you, systems whose workings are secret are far more vulnerable to exploitation than those which are not; this is because the more people who know how the system works, the more chance there is of them spotting errors, mistakes or other problems in the software. It's obviously very important that systems designed to catch people lying work properly: no one wants criminals getting away with it on the one hand, or innocent people being harassed on the other.
Unbelievably, James Plaskitt, the Parliamentary Under-Secretary with responsibility for Housing and Council tax benefit in the Department of Work and Pensions chose to beg the question when interviewed for Moneybox. What follows is a paraphrase of the exchange with the interviewer:
Interviewer: What scientific research is there to show that this technology works?
Minister: That's why we're running the pilot schemes, to see if it works.
Interviewer: But that isn't scientific research, is it?
Minister: No, but it's not up to us to review the science.
Interviewer: Surely it's important before you implement this system that you know it works. How will you know if you don't look at the science?
Minister: Well that's why we're running the pilot schemes - to test this thing out. What's more, the operators I've talked to who use this system are convinced it's solid!
With the minister responsible for national rollout talking in circles and presenting anecdote as evidence, it's no wonder that a device that has been shown to be no more accurate than flipping a coin can gain such a grip on the minds of politicians and civil servants, all of whom are seeing it as a magic wand to cut fraud.
How can intelligent people not spot that there's no evidence for something, or think that a pilot scheme will establish its efficacy without anything to compare it with? I hope the answer isn't "because they don't understand how science works".
Remember the recent biometrics scandal, where a person wearing a gelatin overlay over their fingers was able to fool a commercial fingerprint detector 80% of the time? The UK government is committed to launching a national ID card scheme based on just such biometric equipment starting in 2009, with about as much evidence that it will work as Digilog have for their VRA software.
The fact is we don't know that VRA technology works, and have no right to believe that it will work based on the evidence seen so far. The government should not be implementing it.
"Harrow Council, and soon the Department of Work and Pensions, could well be detecting and prosecuting fraudsters using software whose only claim to be efficacious is that the company that makes it says it is."
Today I find that this is not just my opinion, but also that of Paul Lewis, presenter of Radio 4's Moneybox programme. He popped up on Breakfast on BBC1 this morning to put the arguments against the credibility of these devices much more succinctly than I have done.
In brief, Voice Risk Analysis (VRA) software claims to detect stress in a person's voice, not lying specifically. In day to day use, the software flags up calls from benefit claimants whose voices betray signs of stress and the call can then be followed up with further checks.
The presenters told Lewis that Harrow Council had saved £110,000 by the use of Voice Stress Analysis: Lewis pointed out that the problem was that they had nothing to compare the method to, how did Harrow Council know for instance, that picking people at random and following up their claims would not generate an equivalent saving? Also, simply knowing that Harrow Council was piloting such a scheme might be enough to put off fraudulent claimants (as well as discouraging genuine claimants wary of harassment).
Superbly put. Excellent piece. Also in Lewis's favour was his mention of Niels Bohr in the context of making predictions about the future. Anyone who can mention a famous physicist at 8:45 in the morning on national television gets my vote.
A piece on VRA also appeared later in the day on the 29th December issue of Moneybox. Lewis was presenting and went into a lot more detail.
It turns out that the supplier of the software is Digilog as predicted on this blog back in September. Digilog, it seems, will not tell anyone how it is able to detect stress in voices, nor why they assume that stress and lying are linked. This should start alarm bells ringing in any reasonable person's head. Contrary to what your intuition might tell you, systems whose workings are secret are far more vulnerable to exploitation than those which are not; this is because the more people who know how the system works, the more chance there is of them spotting errors, mistakes or other problems in the software. It's obviously very important that systems designed to catch people lying work properly: no one wants criminals getting away with it on the one hand, or innocent people being harassed on the other.
Unbelievably, James Plaskitt, the Parliamentary Under-Secretary with responsibility for Housing and Council tax benefit in the Department of Work and Pensions chose to beg the question when interviewed for Moneybox. What follows is a paraphrase of the exchange with the interviewer:
Interviewer: What scientific research is there to show that this technology works?
Minister: That's why we're running the pilot schemes, to see if it works.
Interviewer: But that isn't scientific research, is it?
Minister: No, but it's not up to us to review the science.
Interviewer: Surely it's important before you implement this system that you know it works. How will you know if you don't look at the science?
Minister: Well that's why we're running the pilot schemes - to test this thing out. What's more, the operators I've talked to who use this system are convinced it's solid!
With the minister responsible for national rollout talking in circles and presenting anecdote as evidence, it's no wonder that a device that has been shown to be no more accurate than flipping a coin can gain such a grip on the minds of politicians and civil servants, all of whom are seeing it as a magic wand to cut fraud.
How can intelligent people not spot that there's no evidence for something, or think that a pilot scheme will establish its efficacy without anything to compare it with? I hope the answer isn't "because they don't understand how science works".
Remember the recent biometrics scandal, where a person wearing a gelatin overlay over their fingers was able to fool a commercial fingerprint detector 80% of the time? The UK government is committed to launching a national ID card scheme based on just such biometric equipment starting in 2009, with about as much evidence that it will work as Digilog have for their VRA software.
The fact is we don't know that VRA technology works, and have no right to believe that it will work based on the evidence seen so far. The government should not be implementing it.
Thursday, 27 December 2007
A Secret Santa Game
(Note: originally, this article was written using Mathematica, a computer algebra package. Unfortunately, the conversion from Mathematica's native display format to HTML did not work properly, so the text had to be transferred by copy and paste, and the calculation inputs and outputs had to be done "by hand" where possible, and by conversion to bitmap where not. Apologies for the resulting confused appearance.)
A group of friends take part in a game of Secret Santa at Christmas. There are n people taking part.
What is the probability that m pairs of people will have each other as their Secret Santa?
The way to tackle this question is to find the probability that m particular pairs of people will have each other as their secret Santa, then multiply that probability by the number of possible groups of m pairs. A pair of people that have each other as their Secret Santa is called a reciprocating pair.
First we need to find the probability of finding one matching pair of people in a group of n friends. Let's start by imagining that there are 5 friends taking part: Dave, John, James, Jane and Erica. What's the probability of finding a reciprocating pair?
Well, we need one person to pick someone, who goes on to pick him or her in return:
This is 1/4 x 1/4 = 1/16.
This works because there is a 1 in 4 chance of say Dave picking John (he can't pick himself), and a corresponding 1 in 4 chance of John picking Dave.
In general, for n people, the probability is 1 in (one less than the number of people taking part), multiplied by itself.
This is 1/(n-1)^2
The probability that a further pair will have each other must now be calculated from the remaining people. In our example, we are down to James, Jane and Erica, which means that the number of reciprocating pairs is 3 x 2 = 6. In general though, we now have two fewer people than we did before to choose from. That's n - 2 people. Using the rule that a particular person can't pick him or herself, we can calculate the probability of a second match as
1/((n-2) - 1)^2 or 1/(n - 3)^2
The probability of a third match is calculated using the fact that we now have two fewer people to choose from again, or n - 4 people. This gives us 1/((n-4) - 1)^2 or 1/(n - 5)^2.
The probability of an mth match can be found by glancing at the numbers after n in the denominator of each fraction. You can see that they form a sequence 1,3,5,... The position-to-term rule for a sequence of this type is 2m - 1, where m is the number of the match. This is 1/(n - (2m - 1))^2.
To find the probability of a particular three pairs matching in Secret Santa, you just multiply the probabilities of a first, a second and a third match together.
For a game with 11 people (i.e. n = 11) this works out to 1/230400.
This low figure means that for an eleven person game, the probability of a particular three pairs matching, say Amy - Bill, Charles - Derren, and Emily - Fred is vanishingly small.
What about the probability of a particular m pairs matching? To calculate this, we need the probability of the mth pair matching, which we already know is 1/(n - (2m - 1)^2, and then multiply it by the (m -1)th probability, and the (m -2)th probability, and so on until we come to m = 1.
To do this, we use the product function, which multiplies its terms together in a way analogous to the sum function ∑. Here, the range variable is i.
∏ 1/(n - (2i -1))^2 (evaluated from i = 1 to m)We are half-way to a general formula. We now need to know the number of possible groups of m particular reciprocating pairs chosen from n people. This is easier than it seems. The way to count the number of pairs in n people is n(n-1). Look at the possible pairs for our original 5 people: Dave, John, James, Jane, and Erica.
Dave - JohnDave-James
Dave-Jane
Dave-Erica
John-Dave
John-James
John-Jane
John-Erica
James-Dave
James-John
James-Jane
James-Erica
Jane- Dave
Jane- John
Jane-James
Jane-Erica
Erica-Dave
Erica-John
Erica-James
Erica-Jane
Looking in the left column, you can see that each of the five names appears, and is matched with everyone else except themselves. This means each person is matched with four others. That is why the formula for possible pairs is the number of names multiplied by one less than this number: n x (n - 1). In our particular case of 5 names, there are 5 x 4 = 20 possible pairs. Of course, to get a reciprocating pair Jane-John and John-Jane must be combined - Jane buys for John and John buys for Jane. This means the number of reciprocating pairs is half the number of possible pairs. The formula now becomes (1/2)n(n - 1).
Having matched up two people, they are removed from the group, which now has n - 2 members. We then count the possible pairs again. The formula for the next set of pairs is therefore (1/2)(n - 2)(n - 3).
We have n - 4 players left, so the formula is (1/2)(n - 4)(n - 5) and so on. To find the total number of groups consisting of three reciprocating pairs out of n people, we simply multiply our three formulae together:
(1/2)n x (n - 1) x (1/2)x (n - 2) x (n - 3) x (1/2) x (n - 4)(n - 5). For 11 people this is
(1/2) x 11 x 10 x (1/2) x 9 x 8 x (1/2) x 7 x 6 = 41580.
For n people, we use another product formula:
∏((1/2)
x (n - 2i) x (n - (2i+1)) (evaluated from i = 0 to m - 1)Using Mathematica, we can now combine the formula for m particular matches with the formula for m possible groups out of n people:
This gives us
The Pochhammer function takes inputs n and m and evaluates
according to the ruleIt is related to the factorial function.
Finally, we define a function whimsically called SecretSanta, which takes inputs n and m and gives us the probability of finding m reciprocating pairs in n people.
For n = 11 and m = 3 the output is given below.
SecretSanta[11, 3]
0.180469
Or an 18% chance that there will be three matching reciprocating pairs.
This 3D visualisation shows how the probability of any number of matches falls off quite dramatically with rising numbers of people taking part.
Sunday, 25 November 2007
Cranial Osteopathy
Cranial osteopathy, otherwise known as craniosacral therapy, is a type of osteopathy that involves "manipulation" of the bones of the skull to promote mental and emotional health. It was invented by William Sutherland, an osteopath working in the 1930's.
It is based on the idea that the human brain and hence the cerebro spinal fluid surrounding it pulses rythmically in a way unrelated to heart-rate, that these pulses can be felt with the fingertips, and that illness can be caused by restricting the flow of cerebro-spinal fluid.
Practitioners claim to be able to gently manipulate the bones in the skull in order to relieve flow blockages and cure or alleviate the symptoms of disease. Because the treatment is so gentle, it is promoted as being especially suited to babies and young children.
The Craniosacral Therapy Association of the UK claims that one of the causes of problems in babies and children is "displacement of things in their bodies" caused by compression of the skull during birth. This can lead to all sorts of problems, including behavioural difficulties. Fortunately, most of these problems respond very well to craniosacral therapy, which it describes as "a subtle and profound healing form which assists the body's natural capacity for self-repair."
Stephen Barrett, of the Quackwatch website, says that craniosacral therapy has no therapeutic value. This is for two reasons. One, the underlying theory is demonstrably false: the brain does pulsate, but this is solely for cardiovascular reasons, and no connection has ever been demonstrated between brain pulsation and general health. Two, tests done on practitioners of craniosacral therapy revealed that their examinations of the same twelve patients revealed very different rates of brain pulsation, which Barrett notes are exactly the outcomes one would expect from people trying to measure a non-existent phenomenon.
What does this mean for a mother who is worried about her child's behavioural problems? If craniosacral therapy is as gentle as its proponents claim, there is little harm to be done to the child, only to the mother's pocket. More serious is the possibility that there is something genuinely wrong with the child that your average craniosacral therapist wouldn't spot. Stephen Barrett comments that most such therapists have such poor judgement that they should be delicensed. This is in the USA of course. In the UK, a person working as a craniosacral therapist needs no qualifications at all.
Sunday, 11 November 2007
Genetically Modified Food
The Liberal Democrat party's DEFRA spokesperson has waded into the debate on GM food policy in the UK by commenting on a ministerial statement by the UK Government.
The statement, from Phil Woolas, the minister for the environment, referred to a recent consultation on the coexistence of GM and non-GM crops. The statement simply announced that the government wanted to wait until all the evidence is in before formulating a policy on growing GM crops near conventional crops. The government wants the policy so that the public can be sure that conventional crops are not contaminated by their GM neighbours.
The Liberal democrat spokesman, Chris Huhne , said
"People want to be safe and not sorry on GM foods, as the overwhelming bulk of responses to the Government's consultation show. Ministers should not give any go-ahead for commercial planting until they can state confidently that GM varieties would not contaminate non-GM foods and that they are safe."
Why is it important that GM crops not contaminate their non-GM counterparts, for example by cross-pollination? The answer appears to be that the British public thinks GM crops are probably dangerous, or at least have not been conclusively shown to be safe.
Is there any evidence that GM foods can cause harm to human beings? It appears not. An independent review of over 600 scientific papers in 2003 concluded that
"To date world-wide there have been no verifiable untoward toxic or nutritionally deleterious effects resulting from the cultivation and consumption of products from GM crops. However, absence of readily observable adverse effects does not mean that these can be completely ruled out and there has been no epidemiological monitoring of those consuming GM food."
Consumers in the United States have been eating GM food for more than ten years, and GM foods have been introduced in Canada, India and Australia. Presumably, we can look to these countries for some epidemiological monitoring.
The scientific evidence also seems to show that the negative impact on human health of various manipulations of plant DNA is low. This is unlikely to persuade a British public still aching from the scars of the BSE crisis of nearly twenty years ago, which is the most likely reason for public antipathy towards statements made by governments and scientists about food safety.
Campaigners in this country have exploited the fear that the BSE scare engendered in the public mind to mobilise public opinion to levels unheard of in the rest of the world. Couple this with images of "Frankenstein foods" evoked in the British media and it is no surprise to find that the result has been to hold back British progress in a very promising field of scientific research; to force scientists working in these areas to consider leaving the country to further their careers elsewhere; and to produce nothing of discernable benefit to the British consumer.Of course,one can't blame politicians for wanting to react appropriately to the public mood, but would it be too much to ask that they introduce statements like Chris Huhne's with a caveat such as "There's no evidence that GM foods are harmful to humans"?
Saturday, 3 November 2007
Is organic food better for you?
No less an institution than the BBC recently reported that organic food is "better for you" than the conventionally farmed stuff. You can see the video report here. In it, the reporter says that "organically farmed milk, fruit and vegetables are more nutritious than conventionally farmed produce".
The BBC does not mention here exactly what is meant by "nutritious", but reports that up to 40% more antioxidants could be found in organic fruit and vegetables than in non-organic, and quotes the head of the research team, Professor Carlo Leifert, as saying "We have shown there are more of certain nutritionally desirable compounds and less of the baddies in organic foods, or improved amounts of the fatty acids you want and less of those you don't want."
The claims stem from research being carried out at Nafferton Farm near Newcastle Upon Tyne where a comparison study is being performed between organically and conventionally farmed vegetables. The research is being coordinated by Newcastle University's School of Agriculture, Food and Rural Development. It is funded to the tune of 12 million pounds by the European Union, under their Quality Low-Input Food (QLIF) initiative.
The QLIF initiative is a long-term "integrated project" to improve knowledge of the benefits and drawbacks of organic farming - since most of the consuming public seems increasingly fixated on the subject.
The QLIF initiative maintains its own website and publishes its own scientific papers which are available via an on-line archive set up by an umbrella organisation for research into organic methods and based in Denmark.
Before we jump ship and make the switch to organic however, we should reassure ourselves of two things: one is that we are not making the mistake of assuming that because nutritional quality might be higher in one type of food than another, it's therefore appropriate to switch to the higher quality food. The other is to make sure that the usual, rather basic pre-requisites of science apply: the research the BBC reported on should have been published in a peer-reviewed journal of reputable standing in the field in which the claims are being made, and it should have been repeated, preferably by someone else somewhere else, to ensure that the results found are not artefacts of the experimental design.
The trouble is that hardly any of the scientific papers that appear to relate to the comparisons between organic and conventional foods on the QLIF website have been peer-reviewed. Out of 14 papers on effects of production methods, only two had been peer-reviewed. Of these, one (published in 2007) stated that it was not possible to "draw overall conclusions about the effect of low input production on food quality and safety" unless more research was carried out; the other concerned food that was being fed to rats, and found that the content of lutein was higher in feeds prepared from organic produce. Whilst this second result is certainly positive, it is hardly earth-shattering evidence of benefit to humans.
Also, the BBC report made plain that the results of the current research will not be published until next year. The BBC did not even make clear, in their reports that talked about the higher nutritional content of organic foods, whether they were referring to previous work that had been published, or to the as-yet unpublished results of the current research.
So let's straighten this out. It looks like the BBC has been reporting on unpublished and incomplete research carried out by a team whose leader claims higher nutritional content in organic food, whilst in possession of no peer-reviewed evidence to make such an assertion. Of course, there may be evidence in hard-to-find places, behind an academic pay-wall for instance. On the face of it, however, this seems to leave the question as to whether organic food is better for us as just that: a question. It certainly does not seem to justify the rather positive spin placed on the news by the media. Indeed, one wonders, given that there seems to be nothing new to report, where the media picked up the idea in the first place.
Saturday, 27 October 2007
Contemplative wonder and limitless scientific knowledge - at odds?
The Face to Faith commentary section in today's Guardian contains this article, by Mark Vernon, a priest turned agnostic journalist and author of Atheism: Science, Religion and the Meaning of Life.
Vernon argues that if we view scientific knowledge as being limitless in its scope, we are eroding our ability to engage in "contemplative wonder" at things which science cannot explain.
He draws a distinction between contemplative and "instrumental" wonder: the instrumental variety is the sort of wonder we might feel when solving a puzzle, a wonder that fires a desire to see the puzzle solved; the contemplative variety is the wonder "which does not undo, but lets be" - this type of wonder is the type our ancestors might have felt on seeing an approaching storm: an awestruck sense of connectedness between human beings, the natural world and the divine.
Having no problem with either type of wonder, one may be forgiven for asking what all the fuss is about. It turns out that Vernon is worried that our obsession with the instrumental type of wonder means that nothing is left sacred. There are some things, he asserts, that are beyond the comprehension of science: "consciousness, morality and existence itself" he gives as examples.
Apparently, the artistic, religious and moral imagination are well-equipped to ponder these matters. This leaves one thinking that the effect of Vernon's assertions about the nature of contemplative wonder is to set up a series of intellectual "no-go" areas where science cannot and should not probe.
The examples he gives of consciousness and morality in particular are areas where science is just starting to have the ability to explore. There's no problem with the idea of contemplating something "as it is" without wishing to probe any further, but seeking after a particular area of knowledge should not be prohibited simply because someone thinks it ought to be "sacred".
Auguste Comte said that mankind would never know the nature of the stars shortly before the invention of the spectroscope which yielded a treasure trove of information about their structure. Although he didn't make a moral case for not inquiring as to their nature, where would humanity's knowledge of astronomy be if everybody had believed him and turned their attentions elsewhere?
Vernon's remark about contemplative wonder being something which "does not undo but lets be" sounds a little like Tolkien saying "He who breaks a thing to find out what it is has left the path of wisdom". Although in the case of a computer it might well be a bad idea to trash it in order to find out how it works, in particle physics it is more or less the only way that progress can be made.
If we set up arbitrary barriers to knowledge on the grounds that some things must be sacred, we risk depriving ourselves of a valuable insight into the human condition. A life without meaning is indeed an impoverished one, but there's absolutely no reason why we can't create our own meaning.
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