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Newcomb’s Paradox: A Simple Thought Experiment

A Version of this article is published in my book, ‘TWISTED LOGIC: PUZZLES, PARADOXES, AND BIG QUESTIONS’. CRC Press/Chapman & Hall, 2024https://www.amazon.co.uk/Twisted-Logic-Puzzles-Paradoxes-Questions/dp/1032513349

A Thought Experiment

Newcomb’s Paradox, also known as Newcomb’s Problem, is a thought experiment involving a choice between two boxes: one transparent containing $1,000 and one opaque that may contain nothing or $1 million. The twist? A highly accurate Predictor has already decided what’s in the opaque box based on what it thinks you will choose. It is a dilemma first proposed by William Newcomb, a theoretical physicist, and popularised by philosopher Robert Nozick.

The Setting of Newcomb’s Paradox

In this setting, the simple choice is being taking both boxes or just taking the opaque box. The Predictor, which is well known for its accuracy, determines the content of the opaque box based on a prediction about your decision. If the Predictor forecasts you will take both boxes, it places nothing in the opaque box. On the other hand, if it predicts that you will only take the one box, a sum of $1 million will be deposited inside. By the time you make your decision, the Predictor’s choice about the opaque box’s content has already been made. So, should we take both boxes or just the one opaque box? You could also change the amounts to update to modern day prices or in some other way and ask yourself whether it changes anything.

Two-Boxers vs. One-Boxers: The Great Debate

Essentially, Newcomb’s Paradox has divided people into two distinct camps, each adhering to a different way of looking at the problem. These factions, known as ‘two-boxers’ and ‘one-boxers’, represent different facets of decision-making theory and reflect different approaches to rational choice.

Two-Boxers: The Dominance Principle and Causal Decision Theory

Two-boxers advocate that the most rational decision is to take both boxes. This perspective aligns with the principle of dominance in decision theory, which states that if one action produces a better outcome than another in every possible scenario, then that action should be chosen. In the case of Newcomb’s Paradox, two-boxers argue that the Predictor’s decision—having already determined the content of the opaque box before you choose—cannot be influenced by your subsequent choice. This means that choosing both boxes can’t make you worse off. In the worst-case scenario, you have the guaranteed $1,000 from the transparent box, and in the best-case scenario, you could walk away with an additional $1 million if the Predictor failed in its prediction.

Two-boxers also fundamentally subscribe to causal decision theory. They reason that since your decision doesn’t cause a change in the already-decided contents of the opaque box, it’s only rational to maximise the guaranteed gains, which means taking both boxes. This standpoint portrays the logic of irreversibility, where past events (the Predictor’s decision) cannot be influenced by future actions (your choice).

One-Boxers: Evidential Decision Theory and Trusting the Predictor

Conversely, one-boxers argue for a different interpretation of rationality. They propose that the sensible decision, given the Predictor’s uncanny accuracy, is to take only the opaque box. They reason that, although the contents of the box have been decided, the Predictor’s ability to forecast accurately makes it likely that the opaque box contains the $1 million if you choose it alone.

One-boxers point to the track record: almost every participant who opted for two boxes found the opaque box empty, while the opposite was true for those who took only the opaque box. Hence, they argue that it’s not about changing the past, but about leveraging the evidence that shows a strong correlation between the decision to pick one box and winning the million dollars.

In essence, one-boxers align with evidential decision theory, which suggests that we should make decisions based on the best available evidence for the desired outcome. In the context of Newcomb’s Paradox, taking only the opaque box is based on what has happened in the past to those who took one box and two boxes, respectively.

In this way, the paradox challenges our notions of causality and rational decision-making. Can our current choice affect a decision that’s already been made? Or does the Predictor’s accuracy mean it’s better to trust the pattern of past outcomes?

Split Decision

The paradox thus splits decision-makers into two groups: ‘two-boxers’ and ‘one-boxers’, each advocating for a different decision based on their own logic.

Two-boxers argue that the rational decision is to take both boxes. As the Predictor’s decision about the content of the opaque box is already determined before you choose, your choice can’t change the contents. This implies that no matter what, you won’t be worse off taking both boxes. The least you can get is the $1,000 from the transparent box, and at best, you could get an additional $1 million if the Predictor predicted incorrectly.

On the other hand, one-boxers argue that the sensible decision, considering the Predictor’s near-perfect track record, is to take only the opaque box. They point out that almost everyone who has taken two boxes has found the opaque one empty, while those who took only the opaque box won the million dollars. Thus, based on the evidence, it seems sensible to become a one-boxer.

The decision-making here presents a fascinating conflict between reason (which seemingly lacks supporting evidence) and evidence (which seemingly lacks rational explanation). It essentially raises the question: should we trust the evidence of a well-documented pattern or rely on the rational logic of decision-making?

Causality: The Predictor and the Future

The first crucial point to clarify in Newcomb’s Paradox is the nature of causality at play. The scenario eliminates any notion of backward causality or retro-causality; your choice does not affect the Predictor’s prior decision nor alter the content of the opaque box. This stipulation aligns with our typical understanding of time’s arrow: the future does not influence the past.

The Predictor’s decision is a genuine prediction and doesn’t involve any time-travelling. It infers your choice before you make it, but it doesn’t ‘react’ to your decision.

Predictability: Unravelling the Accuracy of the Predictor

The Predictor’s high accuracy complicates the decision-making process. If you tend to be a two-boxer, you might think that it’s likely the Predictor has foreseen this and left the opaque box empty. Conversely, if you lean towards one-boxing, you might believe that the Predictor has probably predicted this and filled the opaque box with the million dollars.  The paradox then becomes less about the boxes you choose and more of a high-stakes mind game where you try to leverage the Predictor’s uncanny accuracy for your gain.

Identity: The Person You Choose to Be

This leads to another fascinating dimension: the intersection of predictability and identity. If the Predictor can predict your decision based on your inherent nature, then maybe the real strategy lies in manipulating your own disposition to game the system. The question then evolves from ‘which box should you choose?’ to ‘who should you choose to be?’

In essence, if you aspire to secure the million dollars, the optimal strategy might be to become the type of person who would always choose one box. The paradox suggests that by firmly committing to this decision, you make it likely for the Predictor to foresee this choice and fill the opaque box accordingly.

The role of the Predictor, therefore, not only tests our understanding of causality and predictability, but it also nudges us to introspect on the role our identity plays in decision-making. It prompts us to consider the potential power of a self-fulfilling prophecy, where our decision to be a certain ‘type’ of person may indeed lead to the desired outcome. Thus, Newcomb’s Paradox elegantly encapsulates the intricate interplay between causality, predictability, and personal identity in shaping our choices and their consequences.

Conclusion: One Box or Two?

The question remains: why leave the sure $1,000 in the transparent box when the content of the opaque box is already decided? Why not take both? This question is at the heart of Newcomb’s Paradox. The paradox doesn’t necessarily dictate a ‘correct’ decision. Instead, it presents a problem that forces you to rethink rationality, predictability, and decision-making. It also highlights the complexity and paradoxical nature of decision-making when dealing with highly reliable predictors. Ultimately, though, the decision rests with you. Would you take one box or two?

Where are the Aliens?

A Version of this article is published in my book, ‘TWISTED LOGIC: PUZZLES, PARADOXES, AND BIG QUESTIONS’. CRC Press/Chapman & Hall, 2024. https://www.amazon.co.uk/Twisted-Logic-Puzzles-Paradoxes-Questions/dp/1032513349

Where Is Everybody?

In the early 1950s, the world was on the cusp of the Space Age, with rapid advancements in rocketry and a growing fascination with outer space. It was a time of optimism and curiosity about the cosmos, fuelled by science fiction and the nascent space programmes. Enrico Fermi, a Nobel Prize-winning physicist known for his work on the Manhattan Project, posed a question during a casual lunch conversation with colleagues, sparking a debate that would extend far beyond that moment. Fermi’s question ‘Where is everybody?’ resonated deeply. It juxtaposed the era’s technological optimism with a sobering, profound mystery. Given the vastness of the universe, why is there no evidence or contact with any extra-terrestrial civilisations?

The Age and Size of the Universe

The age and size of the universe are key aspects of the Fermi Paradox. The universe is approximately 13.8 billion years old, and the Milky Way galaxy, where our solar system resides, is about 13.6 billion years old. By comparison, the Earth is about 4.5 billion years old. This vast timescale implies that if the evolution of life and development of technological civilisations is a common process, there should have been ample time for numerous advanced civilisations to arise in our galaxy alone.

The sheer size of the universe reinforces this idea. The Milky Way is home to an estimated 100 billion to 400 billion stars, many of which are likely to host their own planetary systems due to the prevalence of elements necessary for planet formation. This gives rise to an incredibly large number of potential sites for life.

Recent astronomical discoveries have further accentuated the perplexity of the Fermi Paradox. The launch of modern telescopes like Kepler and TESS has led to the identification of thousands of exoplanets, many of which are in the habitable zone of their stars, and it has only deepened the enigma of the paradox, making the silence in the cosmos even more confounding.

Technological Advancement and Singularity

The concept of technological singularity—a point where technological growth becomes uncontrollable and irreversible—presents a fascinating intersection with the Fermi Paradox. If other civilisations have reached singularity, leading to exponential growth in their capabilities, why is there no evidence of their existence? This discrepancy raises questions about the nature of advanced civilisations and their technological trajectories. If we consider the rapid pace of human technological development, it’s reasonable to think that an extra-terrestrial civilisation, with a head start of even a few thousand years, would have achieved technological feats beyond our comprehension.

Could it be that the very nature of singularity leads civilisations to evolve in ways that are undetectable to us, or perhaps, that the pursuit of singularity inadvertently leads to self-destruction?

Proposed Solutions

The Zoo Hypothesis is one proposed solution to the Fermi Paradox. It suggests that extra-terrestrial civilisations are aware of our existence but have intentionally chosen not to contact us but perhaps to observe us. This could be due to a policy of non-interference, aimed at allowing younger civilisations like ours to develop and evolve independently.

The Great Filter hypothesis proposes that there is a critical barrier or a series of barriers that drastically reduce the probability of intelligent life arising, persisting, and becoming detectable by others. The concept of the Great Filter helps explain the lack of observed extra-terrestrial civilisations by suggesting that one or more critical steps in the development of life or civilisation are extremely unlikely or have a high probability of self-destruction.

A related hypothesis is the Rare Earth Hypothesis, which suggests that while simple life forms might be relatively common, more complex, multicellular organisms are exceptionally rare.

The Transcension Hypothesis offers a different perspective on the Fermi Paradox. It proposes that advanced civilisations might not expand outwards into the cosmos but rather inwards, by miniaturising and compressing their technological and informational systems. As a civilisation advances, it might focus on developing virtual realities, advanced simulations, and artificial intelligence rather than pursuing interstellar travel and communication.

In summary, the Zoo Hypothesis, the Great Filter and Rare Earth hypotheses, and the Transcension Hypothesis represent proposed solutions to the Fermi Paradox. Each offers a distinct perspective on the current lack of observed evidence of intelligent life beyond Earth.

Conclusion: The Search Goes On

A number of hypotheses have been proposed to attempt to solve the Fermi Paradox. Each offers a distinct perspective on the lack of observed evidence of civilisations or intelligent life beyond our planet. These ‘solutions’ explore possibilities such as intentional non-interference, the existence of insurmountable barriers in the development of complex life or civilisation, and the focus on inward technological advancement. While none of these hypotheses provide a definitive answer to the Fermi Paradox, they contribute to the ongoing discussion and encourage further exploration and research in the search for extra-terrestrial intelligence in our galaxy and beyond.

In conclusion, the Fermi Paradox and its related hypotheses serve not only as scientific enquiries but also as philosophical and ethical touchstones for humanity. They encourage us to ponder our existence, our future, and our responsibilities in the cosmic arena. As we continue to explore the universe and search for answers, it is possible that we may ultimately learn more about ourselves than about the cosmos that surrounds us.

David Henry Morris Williams

This page is devoted to my Dad.

David Henry Morris Williams, C. Eng.

This page is devoted to my Dad.

In the wrong hands, vaccination statistics can prove deadly. Simpson’s Paradox shows why.

There has been much discussion of late about data published on 1 November, 2021, by the Office for National Statistics (ONS). It is titled ‘Deaths involving COVID-19 by vaccination status, England: deaths occurring between 2 January and 24 September 2021’.

https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/deaths/bulletins/deathsinvolvingcovid19byvaccinationstatusengland/deathsoccurringbetween2januaryand24september2021#deaths-by-vaccination-status-england-data

The raw statistics show death rates in England for people aged 10 to 59, listing vaccination status separately. https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/deaths/datasets/deathsbyvaccinationstatusengland

Counter-intuitively, these statistics show that the death rates for the vaccinated in thus age grouping were greater than for the unvaccinated. These numbers have since been heavily promoted and highlighted on social media by anti-vaccine advocates, who use them to argue that vaccination increases the risk of death. 

The claim is strange, though, because we know from efficacy and effectiveness studies that COVID-19 vaccines offer strong protection against severe disease. For example, the efficiency and effectiveness of the Pfizer-BioNTech vaccine has been shown to be well over 90% in this regard in the most recent studies.  https://www.yalemedicine.org/news/covid-19-vaccine-comparison

Vaccine efficacy of 90% means that you have a 90% reduced risk compared to an otherwise similar unvaccinated person, based on controlled randomised trials, while vaccine effectiveness refers to real-world outcomes. On either measure, vaccines work very well indeed.

So, what’s going on here?

Well, closer inspection of the ONS report reveals that over the period of the study, from January to September 2021, the age-adjusted risk of death involving COVID-19 was 32 times greater among unvaccinated people compared to fully vaccinated people. But hold on! How can we square this with the data from the table listing death rates of those aged 10 to 59 by vaccination status?

For the answer we turn to a classic statistical artefact known as Simpson’s Paradox, which seems to pop up and create misleading conclusions all over the place. https://leightonvw.com/2019/02/14/what-is-simpsons-paradox-and-why-it-matters/

It is a consequence of the way that data is presented. 

Essentially, Simpson’s Paradox can arise when observing a feature of a broad, widely drawn group, where there is an uneven distribution of the population within this group, for example by age or vaccination status. Ignorance of the implications of Simpson’s Paradox can generate misleading conclusions, which can be, and in this case are, verydangerous.

The paradox in these particular ONS statistics arises specifically because death rates increase dramatically with age, so that at the very top end of this age band, for example, mortality rates are about 80 times as high as at the very bottom end. A similar pattern is observed between vaccination rates and age. For example, in the 10 to 59 data set more than half of those vaccinated are over the age of 40. 

Those who are in the upper ranges of the wide 10 to 59 age band are, therefore, both more likely to have been vaccinated and also more likely to die if infected with COVID-19 or for any other reason, and vice versa. Age is acting, in the terminology of statistics, as a confounding variable, being positively related to both vaccination rates and death rates. Put another way, you are more likely to die in a given period if you are older and you are also more likely to be vaccinated if you are older. It is age that is driving up death rates not the vaccinations. Without the vaccinations, deaths would be hugely greater from COVID-19. 

So, what if we divide the 10 to 59 group into smaller age groups?

If we break down the band into narrower age ranges, such as 10 to 19, 20 to 29, 30 to 39, 40 to 49, and 50 to 59, we find that the counter-intuitive headline finding immediately disappears. In each age band, the death rates of the vaccinated are vastly lower than those of the unvaccinated. This also applies in the higher age bands – 60 to 69, 70 to 79, and 80 plus.

Basically, unvaccinated people are much younger on average, and therefore less likely to die. 

Yet there are those out there who are more than happy to use these statistics to mislead. The consequence is that many who would otherwise choose to be vaccinated might refuse to do so. In truth, the age-adjusted risk of deaths involving coronavirus (COVID-19) over the first nine months of this year was in fact 32 times greater in the unvaccinated than the fully vaccinated. This is a hugely important statistic, and we must not let statistical manipulation be used to obscure this critical information.The lives of countless people really do depend on us exposing this truth. 

Leighton Vaughan Williams, Professor of Economics and Finance at Nottingham Business School. https://www.ntu.ac.uk/staff-profiles/business/leighton-vaughan-williams

Read more in Leighton’s new publication, Probability, Choice, and Reason. https://www.amazon.co.uk/Probability-Choice-Leighton-Vaughan-Williams-ebook/dp/B09DPTVFFR/ref=sr_1_2?keywords=probability+choice&qid=1638207631&qsid=262-7509985-0691032&sr=8-2&sres=3540542477%2C0367538911%2C1294977482%2C1108713505%2C1138715336%2C0521747384%2C0387715983%2C3030486001%2C1444333429%2CB07KC98Z3C%2C0071381562%2C0631183221%2C0816614407%2C1848722834%2C3319820346%2CB07SZLGZYH&srpt=ABIS_BOOK

When intuition fails, how to use probability, choice, and reason to find the real answers.

Much of our thinking is flawed because it is based on faulty intuition. But by using the framework and tools of probability and statistics, we can overcome this to provide solutions to many real-world problems and paradoxes. Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

Woman waiting at a bus stop

When it comes to situations like waiting for a bus, our intuition is often wrong.

Imagine, there’s a bus that arrives every 30 minutes on average and you arrive at the bus stop with no idea when the last bus left. How long can you expect to wait for the next bus? Intuitively, half of 30 minutes sounds right, but you’d be very lucky to wait only 15 minutes.

Say, for example, that half the time the buses arrive at a 20-minute interval and half the time at a 40-minute interval. The overall average is now 30 minutes. From your point of view, however, it is twice as likely that you’ll turn up during the 40 minutes interval than during the 20 minutes interval.

This is true in every case except when the buses arrive at exact 30-minute intervals. As the dispersion around the average increases, so does the amount by which the expected wait time exceeds the average wait. This is the Inspection Paradox, which states that whenever you “inspect” a process, you are likely to find that things take (or last) longer than their “uninspected” average. What seems like the persistence of bad luck is simply the laws of probability and statistics playing out their natural course.

Once made aware of the paradox, it seems to appear all over the place.

For example, let’s say you want to take a survey of the average class size at a college. Say that the college has class sizes of either 10 or 50, and there are equal numbers of each. So the overall average class size is 30. But in selecting a random student, it is five times more likely that he or she will come from a class of 50 students than of 10 students. So for every one student who replies “10” to your enquiry about their class size, there will be five who answer “50”. The average class size thrown up by your survey is nearer 50, therefore, than 30. So the act of inspecting the class sizes significantly increases the average obtained compared to the true, uninspected average. The only circumstance in which the inspected and uninspected average coincides is when every class size is equal.

We can examine the same paradox within the context of what is known as length-based sampling. For example, when digging up potatoes, why does the fork go through the very large one? Why does the network connection break down during download of the largest file? It is not because you were born unlucky but because these outcomes occur for a greater extension of space or time than the average extension of space or time.

Once you know about the Inspection Paradox, the world and our perception of our place in it are never quite the same again.

Another day you line up at the medical practice to be tested for a virus. The test is 99% accurate and you test positive. Now, what is the chance that you have the virus? The intuitive answer is 99%. But is that right? The information we are given relates to the probability of testing positive given that you have the virus. What we want to know, however, is the probability of having the virus given that you test positive. Common intuition conflates these two probabilities, but they are very different. This is an instance of the Inverse or Prosecutor’s Fallacy.

The significance of the test result depends on the probability that you have the virus before taking the test. This is known as the prior probability. Essentially, we have a competition between how rare the virus is (the base rate) and how rarely the test is wrong. Let’s say there is a 1 in 100 chance, based on local prevalence rates, that you have the virus before taking the test. Now, recall that the test is wrong one time in 100. These two probabilities are equal, so the chance that you have the virus when testing positive is 1 in 2, despite the test being 99% accurate. But what if you are showing symptoms of the virus before being tested? In this case, we should update the prior probability to something higher than the prevalence rate in the tested population. The chance you have the virus when you test positive rises accordingly. We can use Bayes’ Theorem to perform the calculations.

In summary, intuition often lets us down. Still, by applying the methods of probability and statistics, we can defy intuition. We can even resolve what might seem to many the greatest mystery of them all – why we seem so often to find ourselves stuck in the slower lane or queue. Intuitively, we were born unlucky. The logical answer to the Slower Lane Puzzle is that it’s exactly where we should expect to be!

When intuition fails, we can always use probability and statistics to look for the real answers.

Leighton Vaughan Williams, Professor of Economics and Finance at Nottingham Business School. Read more in Leighton’s new publication Probability, Choice and Reason.

How to spot a faker!

In a fascinating article published in the New York Times, Malcolm Browne relates how Dr. Theodore Hill would ask his mathematics students to go home and either toss a coin 200 times and record the results, or else pretend that they had done so. Either way, he would ask them to produce for him the results of their (real or imaginary) coin-tossing experiment.

Dr. Hill’s purpose in this experiment was to show just how difficult it is to fake data convincingly. It just isn’t that easy to make up a random sequence. Based on this knowledge, he would astound his students by almost unerringly picking out the fakers from the tossers!

One of the ways he would do this would be to spot how many times heads or tails would be listed six or more times in a row. In real life, this occurrence is overwhelmingly probable in 200 coin throws. To most of his students this long a sequence is counter-intuitive, an example of what is often termed the Gamblers’ Fallacy, i.e. the erroneous perception that independent random sequences will balance out over time, so that for example an extended sequence of heads is more likely to be followed by a tail than a head. The fakers, susceptible to the Fallacy, are thus easily exposed. Ordinary people, even mathematics students, simply can’t help introducing patterns into what is random noise.

This is an example of a broader analysis which is usually referred to a Benford’s Law, which essentially states that if we randomly select a number from a table of real-life data, the probability that the first digit will be one particular number is significantly different to it being a different number. For example, the probability that the first digit will be a ‘1’ is about 30%, rather than the intuitive 10%, which assumes that all digits are equally likely. In particular, Benford’s Law applies to the distribution of leading and trailing digits in naturally occurring phenomena, such as the population of different countries or the heights of mountains. For example, choose a paper with a lot of numbers and circle the numbers that occur naturally, such as stock prices. So lengths of rivers lakes could be included, but not artificial numbers like telephone numbers. 30% or so of these numbers will start with a 1, and it doesn’t matter what units they are in. So the lengths of rivers could be denominated in kilometres, miles, feet, centimetres, without it making a difference to the distribution frequency of the digits. 

The empirical support for this proportion can be traced to the man after whom the Law is named, physicist Dr. Frank Benford, in a paper he published in 1938, called ‘The Law of Anomalous Numbers’. In that paper he examined 20,229 sets of numbers, as diverse as baseball statistics, the areas of rivers, numbers in magazine articles and so forth, confirming the 30% rule for number 1. For information, the chance of throwing up a ‘2’ as first digit is 17.6%, and of a ‘9’ just 4.6%. The same principle applies to trailing (i.e. last) digits. It’s a great way, therefore, of checking the veracity of receipts. If, for example, there is an unusual number of trailing digit ‘7’s, there’s a decent chance that the figures are cooked.

To explain the basis of Benford’s Law, take £1 as a base. Assume this now grows at 10% per day.

£1.10, £1.21, £1.33, £1.46, £1.61, £1.77, £1.94, £2.14, £2.35, £2.59, £2.85, £3.13, £3.45, £3.80, £4.18, £4.59, £5.05, £5.56, £6.11, £6.72, £7.40, £8.14, £8.95, £9.84, £10.83, £11.92, £13.11, £14.42, £15.86, £17.45, £19.19, £21.11, £23.22, £25.50, £28.10, £30.91, £34.00, £37.40, £41.14, £45.26, £49.79, £54.74, £60.24, £72.89, £80.18, £88.20, £97.02 …

So we see that the numbers stay a long time in the teens, less in the 20s, and so on through the 90s, and this pattern continues through three digits and so forth. Benford noticed that the probability that a number starts with n = log (n+1) – log (n).

NB log10 1 = 0; log10 2 = 0.301; log10 3 = 0.4771 … log10 10 = 1.

Leading digit                                                        Probability

      1                                                                 30.1%

      2                                                                 17.6%

      3                                                                 12.5%

      4                                                                 9.7%

      5                                                                 7.9%

      6                                                                 6.7%

      7                                                                 5.8%

      8                                                                 5.1%

      9                                                                 4.6%

 

Tax authorities are alert to this, or should be, which should make fraudulent activity just that little bit easier to detect, especially when the fraudster is unaware of the Benford distribution. For all right-minded citizens, we can call that Benford’s Bonus.

Links:

http://www.rexswain.com/benford.html

http://www.jstor.org/pss/984802

 

How to distinguish random sequences from contrived sequences.

Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

Ask someone to toss a fair coin 32 times. Which of the following rows of coin toss patterns is more likely to result if they actually do toss the coins and record them accurately, and which is likely to be the fake?

HTTHTHTTHHTHTHHTTTHTHTTHTHHTTHHT

OR

HTTHTHTTTTTHTHTTHHHHTTHTHTHHTHHT

 In both cases, there are 15 heads and 17 tails.

But would we expect a run (r) of five Heads or a run of five tails in the series, where r is the length of the run?

The chance of five heads = (1/2) to the power of r = (1/2) to the power of 5 = 1/32. But there are 28 opportunities for a run of five heads in 32 tosses. Same for a run of five tails.

A good rule of thumb is that when N (the number of opportunities for a run to take place) x (1/2 to the power of r) equals 1, it is likely that a run of length, r, will appear in the sequence. So, a run of length r is likely to appear when N = 2 to the power of r.

In the case of 32 coin tosses, with 28 possible runs of length five, N (28) is almost equal to 2 to the power of 5 (32). So a run of five heads (or of tails) is likely if a fair coin is tossed randomly 32 times in a row, and a run of four is almost certain.

Now look at the series of coin tosses above. The first series of 32 coin tosses has no run of heads (or tails) longer than three. The second series has a run of five tails and of four heads.

It is very likely indeed, therefore, that the second series is the genuine one, and the first one is the fake.

Appendix

Probability of 5 heads in a row = 1/32.

Probability of NOT getting 5 heads in a row from a particular run of 5 coin tosses = 31/32

Chance of NOT getting 5 heads in a row from 28 runs of five coin tosses = (31/32) to the power of 28 = 41.1%.

Therefore, the probability of getting 5 heads in a row from 28 runs of five coin tosses = 58.9%.

Similarly for tails.

The Probability of 5 heads OR 5 tails in a row = 1/32 + 1/32 = 1/16

Probability of NOT getting 5 heads OR 5 tails in a row from a particular run of 5 coin tosses = 15/16

Chance of NOT getting 5 heads OR 5 tails in a row from 28 runs of five coin tosses = (15/16) to the power of 28 =16.4%.

Therefore, the probability of getting 5 heads OR 5 tails in a row from 28 runs of five coin tosses = 83.6%

Probability of 4 heads in a row = 1/16.

Probability of NOT getting 4 heads in a row from a particular run of 4 coin tosses = 15/16

Chance of NOT getting 4 heads in a row from 29 runs of four coin tosses = (15/16) to the power of 29 = 15.4%.

Therefore, the probability of getting 5 heads in a row from 28 runs of five coin tosses = 84.6%.

Similarly for tails.

Probability of 4 heads OR 4 tails in a row = 1/16 + 1/16 = 1/8

Probability of NOT getting 4 heads OR 4 tails in a row from a particular run of 4 coin tosses = 7/8

Chance of NOT getting 4 heads OR 4 tails in a row from 29 runs of four coin tosses = (7/8) to the power of 29 = 2.1%

Therefore, the probability of getting 4 heads OR 4 tails in a row from 29 runs of four coin tosses = 97.9%

Exercise

When Nasser Hussain was England cricket captain during 200-01, he lost all 14 coin tosses in the international matches he captained. Given that he captained England in all international matches about a hundred times, what was the probability that he would face this long a losing streak during his captaincy?

The Problem with Teddy – A Short Story

The Problem with Teddy – A Short Story

By Leighton Vaughan Williams

It was difficult to argue with Teddy. Whether he was right or wrong, I could never be sure. But I was sure of one thing, that if he said something was going to happen, it did. That was the problem with Teddy. He never got it wrong. And soon that would become a problem for me.

 

Chapter 1
The problem with language

A dog can expect its master. That is certainly true. But it can’t expect its master next Tuesday,” said Teddy. “Why not?” I asked. “Because a dog has no concept of time?” “No,” responded Teddy, “it is because a dog has no concept of language.” “So can a lion expect a meal when it sees its wounded prey?” I enquired. “You could ask it,” he said, “but you would never understand the answer. Because even if a lion had language, it would be no language we could ever understand.”

“You see,” said Teddy, “language is how we experience the world, as well as the way that we choose to represent it.”

“So language represents the boundaries of what we can know?”, I asked. “You have said it,” he exclaimed. “In clear, plain language.” “This doesn’t mean that nothing exists that can’t be expressed in language, only that it is outside the limits of our philosophy. “There are more things in heaven and earth than are dreamt of in your philosophy,” I offered. “Yes, than in all our philosophies”, he assured me. “But we can never use philosophy to find or explain them.”

“Can you be sure of that?”, I asked. “There is no way to verify that. And if a statement can’t be verified it is meaningless. That’s the test of a meaningful statement.” For a brief moment I felt clever.

“Why do you say that an unverifiable statement is meaningless?’” Teddy asked me. “In that case your own statement is meaningless.”

It was difficult to argue with Teddy. Whether he was right or wrong, I could never be sure. But I was sure of one thing, that I had never shown him to be wrong. Not to myself or to anyone else. That was the genius of Teddy, but it was also the problem.

 

Chapter 2
The problem with probability

“Even a double-headed coin can come down tails,” said Teddy when he entered our shared workspace, displaying his particularly sprightly gait. And his tap, tap, tap of ebony stick. Now, he didn’t need the walking stick. But he did like to tap, tap, tap it along the floor. That was another problem with Teddy.

“I don’t see how a double-headed coin can come down tails,” came my instant riposte. “It’s all about probability,” he said. “It’s a very low probability, but in the quantum universe, a double-headed coin can definitely come down tails.” I assumed he was right, but I couldn’t see how.

He read a lot, and was proud of what he’d learned. “The man who DOES NOT read the great thinkers has no advantage over the man who CAN NOT read them,” Teddy once told me. “The same goes for a woman,” I said, trying to sound enlightened. I liked to sound enlightened in front of Teddy. I don’t know why. I never did. Even when I said something that I thought made some kind of sense. A glance from Teddy always made that abundantly clear.

But I did admire Teddy’s uncanny ability to distinguish what was going to happen from what was not. He had the gift of what some call prescient foresight but what others might call knowing a sure thing.

You see, when Teddy said something would happen, it happened. Like when he called double-six on the pair of dice I had brought from home. That’s a 35 to 1 chance, logic told me, but in my belly I knew it would happen. I knew that as soon as Teddy said six-six. And six-six it was. I guess you could call it a trick, or you could call it magic. I don’t know about that, but I did now something for sure. If Teddy said it would happen, it would. Never bet against Teddy. That was my watchword. Until I did.

 

Chapter 3
The problem with wagers

“It’s usually best to back the favourite”, I told Teddy. “I had read it in a book. A book by an expert.” “That’s true if you’re talking probabilities,” said Teddy. So now I knew it was true. “But if you know something is going to happen, that doesn’t apply,” he said.

And so we went on Sunday night to the casino, at the insistence of the man who knew when things would happen. We never met at the weekend, but today there was a reason, said Teddy. He knew I would win.

“Let’s play roulette,” he said. And produced a wad of notes, a very big wad of notes. “”Red or black,” he asked. “You choose.”

I chose black. “I would choose red,” he said. “It’s your money,” I said. “No, it’s not,” he replied. “It’s yours now, a thousand pounds, to lose, to double, or to keep.”

“Can I just keep the thousand pounds?” I asked, and not risk it on red, or black. It was a joke, of a kind. Teddy was not a generous man, and certainly not generous enough to gift me a grand. And to me it was a lot of money, money I needed to live.

“You’ve struck lucky in the quantum world,” said Teddy. “The thousand is yours. To keep or to spin. I say red, and I say it’s a sure thing.”

“One spin of the wheel, for the lot, or take it home. Your choice.”

Now, when Teddy said something would happen, it did. And he was saying it was going to be red. But my common sense told me that Teddy could not know. The wheel had not yet even started to spin.

“I’ll keep it,” I declared. A thousand pounds. “OK, cash it in,” he said. “It’s yours.” I protested – what if we share it, I said? But he declined. Teddy didn’t need the money. Knowing what would happen had already made him a rich man. And he was not the kind to share. “Good night,” he said, and tap, tap, tapped off into the gathering twilight.

So to the next day, and I asked him how he knew the ball would have landed on red. “I knew we’d never find out,” said Teddy. “Because I knew you’d never wager a thousand pounds on the spin of the wheel.” “But what if I had spun the wheel?” I asked. “Then you would have won,” he said. “A universe in which you would spin that wheel is a universe in which you would be sure to win.” I thought I understood what he meant.

 

Chapter 4
The problem with money

“Does it make you happy, knowing what’s going to happen?” I asked. “Isn’t it a burden?” “I don’t always know what’s going to happen,” he corrected me. “But when I know for sure that something will happen, it does,” he said. “It’s not at all the same thing.”

“But that’s enough to make you a lot of money,” I said. “Knowing some things for sure that others think are unsure has made you so much money.”

And so he told me the tale of Thales, the Greek philosopher, who made his fortune by the application of modern day principles of analysis to ancient day Greece. The story involved forecasts and finance and options on olive presses. I honestly can’t recall all the details. But Teddy could. “Which shows,” he concluded, “that it is much easier for a philosopher to become rich than for a rich man to become a philosopher. But the ambitions of philosophers are of another kind.” It was clear he was talking about himself.

As for me, I just wanted to be rich like Teddy. I knew I would never be as wise.

But all of his great knowledge, great insight, great wisdom – was a burden to him? He seemed to read my mind.

“Great wisdom does not necessarily bring great happiness,” was his now detached observation. “Nor does great riches.”

“So maybe I’m better off being ignorant old me,” I said. “Just seeking the simple things in life, and enough money to enjoy them.”

He shook his head now, disapprovingly.

“Which is better?”, he asked me, “to be a human being dissatisfied or a pig satisfied, to be Socrates dissatisfied or a fool satisfied?” He was quoting one of the great philosophers again. I could tell that by the way he spoke his syllables. But I didn’t really understand the question, let alone the answer. That, I am afraid, was the problem with me.

 

Chapter 5
The problem with cars

We shared coffee and lunch that day, accompanied by the walking stick, the shiny ebony walking stick. I plucked up courage to ask him about the walking stick, why it accompanied him wherever he walked. “This is not a walking stick,” he replied. I did not ask again.

“So what if I told you that I am sure you will be knocked down by a car tomorrow?”, he now asked me.

“You can’t be sure of that,” I said. “I might not go anywhere near a car.” I suspected he was joking. Not a pretty joke, but Teddy and good taste didn’t always see eye to eye.

He reminded me that there was no way of reaching the office without crossing a road. “I’ll be extra careful,” I said.

“You will be knocked down by a car tomorrow,” he repeated, ” and you will be crippled for life.”

He was deadly serious and now I was scared, because when Teddy knew that something was going to happen, it always did.

“It can’t be inevitable,” I said. “What if I don’t even step outside my front door?” “You won’t do that,” he replied. “You are too curious to see if I’m right.” “Nobody’s that curious,” was my instant response. But I was, because I couldn’t see how he could know this. It was like predicting where the roulette ball would land before the wheel even started spinning. I told him so. “Or like predicting six-six on the dice,” he said. I shuddered – and suddenly felt cold.

How could he know? Had he heard of a plot to harm me? Did he know people who knew? Or was he planning to harm me himself. But if so, why warn me? I could make no sense of the problem, no way through the maze. What would Socrates make of this, I wondered. And what advice would he have for the fool?

I asked Teddy for evidence, for proof. He offered none. He said he knew but said he could not explain. Not to me. He gave no reason, but this told me nothing, because he never did. He never told me how he knew that something would happen, but I knew that it always did.

I turned to close friends, close family. Ignore it. Play safe. He’s just trying to frighten you. Maybe he knows something. A mix of opinions, but nothing to help. Not one of them knew Teddy, nor his ebony stick. And not one of them knew that when Teddy knew something, he knew it for sure.

That was the problem with Teddy. And now it had become a very real problem for me.

Chapter 6
The problem with fate

The day wore on and soon a decision had to be made. A choice to make. A choice between the evidence of my experience, that Teddy was never wrong, or my experience of the evidence, of which there was none. I asked Teddy one last time before we retired to our separate homes. Should I stay home all the next day, or should I brave life’s fate? Could I change destiny?

“All fates are possible,” said Teddy, “but the universe where you will come to no harm is not the universe in which you currently live.” I was thinking back now to that spin of the roulette wheel. In a universe where I spun the wheel, I felt sure I would have won. I chose not to. But I could have done. Surely this meant that life’s events were not pre-destined, written in stone and waiting to simply unfold. I could do something about it. I could have spun that wheel. But that would have been a different universe, where everything would be different. Would it even be me on that universe? I wanted to go back, to ask myself to spin that wheel. But I could never meet myself, because yesterday I was a different person, as are we all. We can never go back and meet ourselves, only meet ghostly shadows of who we were, shadows that made us what we are and who we might have been.

I no longer saw things as they were, asking why. I saw things now as they might be, asking why not.

“I can change the world,” I told Teddy. “I can spin that wheel.”

“Yes, we can change our destinies,” he said. “We have the freedom of will to choose.”

It was approaching six and the caretakers came to shut up the building. It was not the perfect arrangement, but it suited us.

He picked up his ebony stick and set off, with his usual jaunty gait. “You are quite the philosopher now,” he called back, “I’ll see you the day after tomorrow.”

“But …” I started to say. He was gone already. That was the problem with Teddy. Always too quick on that stick.

Chapter 7
The problem with thinking

I woke up at dawn next morning and thought of the double-headed coin that might come down tails. But I knew that I could do nothing about that. The quantum world was out of my control.

But some things were within my control, and one was the choice of whether to change life’s plan, to spin the wheel, to change the course of fate.

This could mean staying home, behind closed doors, away from the rush of traffic. This is what it meant to Teddy. But this is not what it meant to me.

Teddy saw things as they were, and he saw things that would be. I now saw things differently. I saw a world as it might be. Where I had the choice to use reason and faith and hope. To conquer fear, on my own terms.

But reason told me that Teddy’s foresight of my fate was not to be overlooked lightly. Teddy didn’t make that kind of mistake.

But Teddy’s universe wasn’t the one I had to inhabit. I could change my destiny. I could stay home, shuttering out the summer day. But I was becoming a philosopher. And the ambitions of philosophers are of another kind.

“A dog can expect its master, but it cannot expect its master next Tuesday,” Teddy had once explained. I thought of that now as I realised that Teddy was not expecting me today. I had become a philosopher, a thinker. Teddy would soon see.

So I called a taxi, all the way to my front door, and asked to be dropped off at the back entrance to our shared workplace. No cars to knock me down. I would be straight into that taxi, approached from the back. I would ask for the back door of the taxi to adjoin the back door of the workplace. I would give an excuse. Security. And the same when I returned home. Reason over fear. No room for error.

Until the taxi, en route from home to work, came to a halt. On the busy dual carriageway. Something rattling. So Teddy was right. Terrifyingly right. Could I get out and help him identify the noise, asked the driver. No, no, no, I screamed! He looked at me as if I was slightly mad. But this madness had method. To spin the wheel, to save life and limb.

And soon we got going again, me firmly in back seat.

So it was with some surprise, and my almost crazed relief, that we arrived at the door. To park with back door adjoining back door came as a curiosity to the driver. But he nodded sympathetically and I tipped him in thanks.

I skipped up the steps to our plush interconnecting offices, where Teddy wrote software, and I helped him do it. He heard my steps and tried to shut the door, but I was through first. “How are you here?” he shouted. “You’re at home!” Evidently not, I might have replied. Instead, I just stood there, in openmouthed shock at the scene that unfolded before me.

 

Chapter 8
The problem with Teddy

Every drawer had been emptied, every cupboard laid bare, ornaments and accessories opened or turned upside down. If something had been hidden, it would by now have been found. “What is happening?” I would have sat down, but the seats were upturned, and I had no stomach to right them.

“A burglary,” he said. But I didn’t believe him. “Why would burglars turn everything upside down and take nothing?” I asked. “That beggars belief.”

“I disturbed them,” he said, “took about them with my stick. They fled.”

“Let’s call the police,” I insisted, “Check the CCTV.” “No,” he said sharply. “Let’s not.”

A short pause. “Is it safe?” he asked. “Is it safe?”

“Is what safe, Teddy, is what safe?”

He seemed unsure now, what to say or do. “They were my numbers,” he said, “I suggested the numbers. They came up on Saturday night. I know that you keep it here, you always keep your ticket here until you check the numbers on a Wednesday. And I know you never sign it. Be fair, Charlie, let’s share it.”

He looked at me menacingly. Teddy, I knew, was not the sort of man who shared anything. It was all about Teddy. The gift of the thousand pounds now made sense. He had made his case, that I should spin the wheel, that I could re-arrange fate. But a gift so generous. Now I saw. It was his back-up plan.

“No, Teddy, it isn’t safe. I didn’t buy a ticket last week. There’s nothing to share.”

Teddy lunged at me, screaming, before collapsing to the floor, thrashing around. Yet still looking up at me, the look of sheer menace still etched on his face.

I was relieved that I hadn’t bought a ticket. He would have found it, signed it, cashed it, if it had existed. I would not have seen a penny. He had suggested some numbers, but for once this was blind chance. He had not seen the future, the future had grasped him invitingly by the hand. Or so he had thought. And now he sought control, control of what was to come.

I peered now yet further into his soul, and saw it for what it was. I had glimpsed it before. But what I saw now was yet darker. Consciousness without conscience. A man with no love for anything higher or other than himself.

And I saw now how the things he forecast always came true. Because he made them come true. Until now. He was the sort of man who would sell shares in cruise liners and then plant an iceberg, if he could.

“But you would have been rich too!” cried the man who was already rich. The man who lived in a mansion and looked down on the homeless. The man who liked to rip up the charity envelope.

“What shall it profit a man if he should gain the whole world, and lose his own soul?” I asked him now. “Answer me, Teddy!” But no answer came from the man who knew when bad things were going to happen, who knew because he made those things so.

I picked up his stick. I wanted to hit him, to beat him with that shiny, ebony stick. He cowered. A coward, infused with consciousness, but devoid of conscience. I put it down again. It would have given me satisfaction. But it would have made me more like Teddy. For Teddy, his own personal satisfaction was all that mattered.

That was the problem with Teddy. I didn’t want it to be the problem with me.

I sat on the floor, and considered my options.

“I have something to report,” I told the operator. About some bad things that have happened, some things unexplained. Can I speak to the police?”

Bertie’s Big Idea – A Short Story

Bertie’s Big Idea – A Short Story

By Leighton Vaughan Williams

 

Albert ‘Bertie’ Simpson Sinclair was a man who in earlier days might have been described as a bounder and a cad, albeit an immensely likeable and charming member of that sub-species. The problem for Bertie was that he was, as such, a hopeless, if heroic, failure. But Bertie was an optimist, a man who believed in the philosophy of ‘one more push’, of the sure triumph of unsound hope over all too sound experience. And he had an idea which he believed would make him rich. This is the story of Bertie and his magnificent idea.

 

Bertie’s Dream

Mr. Bertie Simpson Sinclair liked to think of himself as an ideas man. And an ideas man he certainly was. He had plenty of ideas, albeit none of them good. But his latest idea was going to be different. Of that he was sure. He had envisaged, in one giant midsummer night’s dream, a scheme to make himself rich, without making others commensurately poor. To this extent, it was an unusual idea for Bertie, for whom all previous schemes consisted of persuading others to part with their money in pursuit of an apparent though negative actual benefit. Bertie called such schemes win-win. By this he meant that he would win twice, first by taking their money, then by virtue of the scheme into which they had invested. The problem for Bertie was that every such scheme remained a dream, for all his boundless wit and charm. Even his plan to sell tips on the horses, then persuade his followers to place their own money on these gems of advice and share with him half the winnings, but none of the losses, failed when faced with the cold light of reality. There were so many others, including Bertie’s ‘Grow rich while you sleep’ manual, his ‘Learn while you doze’ method, his ‘Snooze yourself slim’ prospectus, his ‘Succeed while you slumber’ pamphlet. Bertie reasoned that alert, wakeful people were out of his reach, which left the more reposed segment of the population as his natural target audience. It was not just the fact that he himself was neither rich, learned, successful nor svelte. The real problem for Bertie was that he had singularly failed to convince even one other member of the human race that he could help them become what he so evidently was not. But that, decided Bertie, was about to change. Because of his midsummer night’s dream.

 

Bertie’s Idea

Bertie liked to think of himself as a clubbable man, a sociable ‘bon viveur’ who could mix with natural ease and grace with ladies and gentlemen of refinement. To this end he sought membership of tennis clubs, golf clubs, health clubs, focusing on the most exclusive of each. But Bertie had not grown rich while he slept. On the contrary, he had grown increasingly poor even as he dreamed of growing rich. As such, he was unable to actually gain entry to any of these clubs of the clubbable, as he thought of them. It was all an unrequited dream. But then came the big dream, that midsummer night, the night that inspired Bertie’s big idea. He had dreamed that he was at the door of one of these desirable clubs of the clubbable, begging inwardly to be allowed in, when an elegantly attired gentleman, upon exiting, had spotted the less than svelte figure of the unlearned though charming Bertie, and spoken to him, softly.  “Sir,” he had quietly ventured, “what are you doing waiting at the door? Did you not know that this is a club reserved only for the clubbable?” Taking immediate offence, Bertie’s dreamworld person had risen quickly to his own defence. “I AM a clubbable man,” he had expostulated, invoking his own claims to that most cherished status in society. But something within him had turned, something that was stirred by the well-dressed accuser. And so awoke Bertie, with his brand new big idea, an idea which he had instantly concluded would make him rich.

 

Bertie’s plan

A club for the unclubbable! That’s what he would create. He would create the world’s first club which would only accept members who didn’t want to join, members who were truly unclubbable. He would in other words create a club for those unwilling to join any club that would accept them as a member. The idea was one thing, turning it into a practical scheme was quite another. But that, for Bertie, was the challenge. And the rewards beckoned for Bertie like a shining beacon on a golden hilltop. At least that’s the way that Bertie visualized things. But he knew he was at base camp and the climb that lay ahead was steep and possibly long.

He was not a gifted thinker, but he did have thoughts, and the first of these was to place an advertisement in the local newspaper. Although a man of strictly limited means, it was his only hope of starting the climb which would take him to that shining beacon atop the golden hilltop. The advert was quite simply stated. “Would you join any club that would accept you as a member? If so, we’re wrong for you. We are the world’s only club for the unclubbable. We accept all and only those who don’t wish to join us.”

It was more words than Bertie could really afford, but he had seen that beacon atop the glittering hill and this was his one-time chance to glimpse its light. In the face of that shining lamp, he was steadfast. He would not blink. He waited. For the first response. It arrived by mail the very next day. Addressed to Mr. A. Sinclair, the envelope contained one sheet of blue vellum notepaper. In neat lettering, it was from a Mr. Charles Bone, who simply enquired whether there was an active membership of the club. If so, he was not interested. If not, he might be. Bertie replied with alacrity.  “There is no active membership, so we do not wish to accept you as a member.” By return of post, Mr. Bone accepted membership of this club that didn’t wish to accept him as a member, on one condition. “I am not an active man, and have no wish to be involved with active people. I will join on this condition,” wrote the first and thus far only member of the world’s first club for the unclubbable.

By the same post came an enquiry from a Miss Edith Spratt, who declared herself unwilling to join the club because, while she had been told of the advert, she was not from the local area. As such, she could not make use of its services, even if she wished to, which she did not. Bertie was delighted to accept her as a member, because she was so clearly unable and unwilling to benefit from membership. He wrote to tell her so. On this basis, Miss Spratt became the second member of Sinclair’s club for the unclubbable.

No fee was asked, and none given, by either Mr. Bone or Miss Spratt. But they served their purpose. Neither could in any way reasonably be classed as active members of the fledgling club, but there was now at least a club in existence, and in their different ways both of its members were of the unclubbable kind. There were no further replies to the advertisement, but Bertie was not discouraged. He had left base camp and set forth up the golden hill. He would not turn back.

And so came to Bertie his next idea. If he could introduce Mr. Bone to Miss Spratt, they might help him spread the word through what he conceived as some form of human chain letter that would spread forth and gather together the great unclubbable hordes, brought together into one vast club composed of only those unable and unwilling to join a club.

“Do you possess transport?” Bertie now wrote to Miss Spratt. “Yes”, came the one word reply. Seizing upon this positive news, Bertie devised a plan to bring together the only members of his brand new club. He offered, though he could ill afford it, to pay the cost of fuel for what would be a 70 mile journey for Miss Spratt. The response from Miss Spratt was quick in coming and even quicker in its message. “Dear Mr. Sinclair, my transport is an electric wheelchair. Yours sincerely, Miss E. Spratt”.

To Bertie, that hilltop was starting to look further away than ever.

 

Bertie’s vision

Was Bertie’s vision turning into a mirage? It was a question that might have deterred many, but not a question that deterred Bertie. If Miss Spratt could not come to Mr. Bone, then Mr. Bone must be brought to Miss Spratt, reasoned Bertie with impeccable rigour. Without further ado, he grabbed his quill-like pen, and rushed off a letter. “Dear Mr. Bone, I would like you to meet Miss Edith Spratt.  Like you, she is totally unsuited to the life of a club. In short, she is totally unclubbable. Yet she is a member of the club to which you belong. I think this remarkable coincidence is too great to be overlooked. For that reason, I would like you to meet Miss Spratt. She lives some distance away, but this has the advantage of offering you a pleasant journey even if the meeting is less pleasant than might reasonably be hoped. I hope you reply affirmatively. Yours sincerely, Albert Simpson Sinclair.”

Mr. Bone responded immediately, posing just one short question. “Is Miss Spratt an active member of the club?” Bertie was eager to re-assure. “No, Miss Spratt is not an active member of the club. I trust this reassures you.” It did. The following day, Bertie received the acceptance of his invitation. All that remained was to persuade Edith Spratt to accept the same invitation to meet Mr. Bone. “Dear Miss Spratt,” wrote Bertie, “I would like you to meet Mr. Charles Bone. He is not a clubbable man, and by natural inclination not an active man, but he shares with you membership of the club which I am proud to manage. I trust this remarkable coincidence offers sufficient grounds for you to accept this invitation. Yours sincerely, Albert ‘Please call me Bertie’ Simpson Sinclair.

The letter of response arrived by return of post. Addressed to Mr. Bertie Sinclair, and written in exquisite script, it was simply expressed. “Dear Bertie, I accept your invitation. Please be so kind as to bring Mr. Bone to me. Yours truly, Edith.”

And so was arranged the meeting between Mr. Charles Bone, retired undertaker, and Miss Edith Spratt, lady of leisure, to take place the following Wednesday at the home of Edith Spratt. Thursday and Friday came and went, as did the weekend, but no news leaked out. For several more days, Bertie rushed each morning to pick up the morning mail. But no letter arrived from either Mr. Bone or Miss Spratt. After two weeks had elapsed, which seemed like three months, Bertie reached for his pen and wrote to Mr. Bone. “Dear Mr. Bone, I hope and trust that your meeting with Miss Edith Spratt went well. Perhaps your meeting went so well that you have had little time to write letters. If so, I would be delighted to hear of this happy news, which you might perhaps share much more widely. Yours expectantly, Albert Simpson Sinclair”.

Sooner rather than later a letter arrived, addressed to Mr. A. S. Sinclair.

“Dear Mr. Sinclair,” it read, “Thank you for arranging the meeting between myself and Miss Spratt. You assured me, however, that the lady was not an active member of the club. I cannot agree with your assessment. Could you in future introduce me to one of your less active members? Yours sincerely, Mr. Charles Bone.”

The human chain letter, it seemed, had come apart at the first link.

Bertie took pen to fresh paper, addressed to Miss Edith Spratt.

“Dear Miss Spratt, I understand that no developments arose out of your rendezvous with Mr. Charles Bone, and that you are no longer in contact. Can you confirm my impression? With sincere regards, Albert (Bertie to you) Simpson Sinclair.”

Two days passed, while Bertie fretted. And then it came. The envelope was coloured pink and addressed to Bertie Sinclair. On matching pink notepaper, it simply stated. “Apparently I was too active for the liking of Mr. Bone, or so he told me. Please do, however, feel free to introduce me to someone from your club rather more active than Mr. Bone. Hoping to hear further. Yours in anticipation, Edie.”

 

Bertie’s day

Bertie had lost interest in Mr. Bone, but not in his project. He still possessed the vision of a network of clubs composed entirely of unclubbable people. But the vision was starting, even to Bertie, to flicker a little. His only hope now, he reasoned, lay with Miss Edith Spratt. But he had nobody else to introduce her to, active, inactive or semi-active. Except himself. And so he resolved to visit Miss Spratt at her residence, disguised as a member of his club for the unclubbable. He wrote as follows.

“Dear Edie (if I may), I am sorry to hear that you were too active a member for Mr. Bone. I prefer to see it from a different perspective – that he was not active enough for you. That can easily be remedied. I have on my books a very unclubbable man, who likes his own company, but who I can assure you is a very active member of the club. I will send him to you next Wednesday, if that is convenient. Kindest regards, Bertie.”

Wednesday did prove convenient, and soon a disguised Albert Sinclair, replete with flowing beard, heavy horn-rimmed spectacles and extravagant moustache, was entering the country residence of the wealthy widow newly self-described as Miss Edith Spratt. Introducing himself as Archibald Henry, former solo arctic explorer, he was at once able to tick two boxes, as both a private man and an active man. Miss Spratt was impressed to meet an explorer, less so a former explorer, and even less so a man who had clearly given up the athletic lifestyle at some distant corner in time. They had little in common, so she asked him whether it was cold in the Arctic. Yes, very cold, he said, and there the discussion of his days as an explorer froze. It was only when he spoke of the club that she lit up, asking him whether he had ever met Mr. Bertie Sinclair. She was disappointed to hear he had not, sharing with him her secret crush on this exciting innovator who had created a wonderful club for the unclubbable, and whose charm and good manners flowed out of every word he committed to paper in his delightful letters. She confided in the former explorer how she secretly wished Bertie would visit.

What had he done? This lady of wealth and refinement wanted him, Bertie, and he had entered her life disguised as a hairy arctic explorer. What should he do? Should he discard the disguise and reveal himself, like some sort of superhero, to be the witty, charming man of her dreams? He thought better of it, if only because he wanted more time to think. He bid her farewell and returned the 70 miles to his small suburban bedsit.

He had not spotted the electric wheelchair she had spoken of, but he had been dazzled by the vintage Mercedes sports car gracing her ample driveway. It somehow made her all the more attractive. He slept fitfully that night, rising at dawn to do what he had to.

Drawing from his battered desk the fine stationary he used for only the most important of communications, he applied modern quill to traditional vellum. “Dear Edie,” he wrote, “Mr. Henry, who visited you at my invitation, has contacted me to express his great pleasure at the making of your acquaintance. He tells me, however, that he is not worthy of your notice, and has asked me to convey his great good wishes to you in all you do. Although I am persuaded that I also am not worthy of your notice, I would be happy to follow in the estimable footsteps of our arctic adventurer in order to make your personal acquaintance, should that be your wish. I remain, with the greatest respect, your humble servant. Bertie.”

The next day dragged heavily on Albert Sinclair, as he waited and hoped for a positive reply. He was waiting at the door next day for the arrival of the postman. A quick reply should mean good news, a slow reply worse news, and no reply the worst news of all. The pink envelope arrived at the first opportunity. He opened it gently, hardly daring to read it. “Dear Bertie. I did have some regard for Mr. Archibald Henry, and believed that under his hirsute exterior probably lurked a fine, attractive gentleman. Still, I expect the excess of hair served him well in the cold arctic climate, and he has now grown well accustomed to it. Yes, I would indeed welcome a visit from your fine self. For a man of your considerable talents as gifted entrepreneur, your humility is a further charming sign of the true gentleman that you so clearly show yourself to be. With regards from your friend, Edie.

Bertie could not contain all the excitement shooting through his body. All that stood between him and the wealthy, attractive widow, it appeared, was the removal of his pencil moustache. As such, he would turn up at the elegant doorway, and introduce himself, Albert Simpson Sinclair, to the lady who would clearly not be able to resist his very considerable charms. Wednesday at noon was the agreed time.

 

Bertie’s meeting

She was waiting for him at the door, and extended her hand to him in such a way that he was not sure whether she was expecting him to shake it or kiss it. He shook it. “It is a pleasure and a delight to make your acquaintance in person,” he opened. “Tea or coffee,” she asked. “Coffee, please”. “White or black?” As a man who had not had either tea or coffee made for him for quite some time, he was not used to being questioned about his preferences in such detail. “Black, please, with milk,” he said. She looked at him quizzically. “Yes, plenty of milk,” he confirmed. Decaffeinated, please.

There was no conversation while the coffee was prepared, and after it was served, little more. The series of awkward silences, interrupted by sips of caffeinated coffee, was eventually interrupted by the chime of the grandfather clock standing in the corner of the room, alerting them to the fact that it was 12.30. It presented a much-needed natural break.

“I must take my leave,” said Bertie, “I have so much business to attend to.” There was a further moment of silence, while Miss Spratt rose to her feet, pointing accusingly at him. “What have you done with Bertie?” she asked. “Tell me NOW, what have you done with Bertie?” He was sure he had misheard her. “What have I done with WHAT?” he asked.

“What have you done with Bertie?” she persisted, in an increasingly strident tone. “But I AM Bertie!” “You, Sir, are NOT. You are Mr. Archibald Henry, former arctic explorer. Do you really think you could trick me into thinking you were my Bertie by shaving off your formerly abundant facial hair.” “No,” she continued, “Mr. Archibald Henry minus beard, moustache and large-rimmed spectacles is still Mr. Archibald Henry. Now tell me what you have done with Bertie, or I shall call the police to have you arrested.”

“I AM Albert Simpson Sinclair,” he insisted, “Archibald Henry does not exist.” At these words, Edith Spratt reached urgently for the telephone. “So you are now saying that you, Archibald Henry, do not exist, even though you stand right before me. Is this your defence to the charge of abducting Mr. Sinclair, or worse? A defence of insanity.”

Bertie could see his Big Idea unravelling before his eyes, the dream giving way to stone cold reality. Maybe he was insane, to hope that any idea of his could come true, maybe he was insane to still dream that one day he could persuade people that they could grow rich while they slept, succeed while they slumbered, learn while they dozed, slim while they snoozed. Maybe he was insane to believe that he could create a club for the unclubbable. But he was not insane in the way that Edith Spratt thought he was, and certainly not criminally insane.

For perhaps the first time in many years, he now decided upon a plan at odds with every instinct in his bones, a plan to tell the truth.

“It was I, Albert Simpson Sinclair, who came to your home last week disguised as the fictional arctic explorer, Archibald Henry. It is I, Archibald Simpson Sinclair, who stand before you now. I throw myself upon your good graces. I can do no more.” He paused. “Edie,” he half sobbed now, “I am Bertie.”

Edith Spratt said nothing but put down the telephone she had been wielding with increasing menace. “Mr. Sinclair,” she said quietly. “I am not sure whether you are a good man or a bad man, a sound man or an unsound man, and I am not really concerned to find out.” Bertie winced. “But”, she continued, “I do know the difference between a good idea and a bad idea, a sound idea and an unsound idea. And I am rather attracted to your big idea.” “A club for the unclubbable?” piped up Bertie, excitedly. “Quite so,” declared Miss Edith Spratt. “I shall turn this idea into reality, and because I am a lady of honour and refinement, you shall be rewarded with a respectable share in its fortunes. But be assured, Mr. Sinclair, this shall become my vision, the vision of Edith Evadne Spratt.

And so began a new dawn for Mr. Bertie Sinclair. Employed to use his considerable wit and charm to help expand the Spratt chain of clubs for the unclubbable, his big idea had become reality. He knew now that he would never grow rich while he slept, nor succeed while he slumbered, but he would indeed grow rich, by working hard while awake, and he would succeed. But much more importantly, Mr. Archibald Simpson Sinclair had now achieved a station in life which neither money nor worldly success could alone bestow. Bertie Sinclair, one-time conman, cad and bounder, had been transformed. Eminently clubbable, he had finally become a gentleman.