A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams, Chapman & Hall/CRC Press. 2024.
Card Counting: A Winning Strategy in Blackjack
In 1962, Ed Thorp introduced a strategy that would forever change the landscape of blackjack: card counting. His book, Beat the Dealer: A Winning Strategy for the Game of Twenty-One, presented a system based on probability theory that allowed players to gain an advantage over the house. Since then, card counting has become a topic of fascination for blackjack players worldwide.
Understanding the Basics of Blackjack
To grasp the significance of card counting, it’s essential to understand the fundamentals of blackjack. The basic objective of the game is simple: players aim to draw cards that beat the dealer’s hand without exceeding a total of 21. While basic strategy provides players with a foundation for optimal gameplay, card counting takes it a step further by incorporating the knowledge of which cards have already been dealt.
The Concept of Card Counting
Card counting revolves around the concept that certain cards have a different impact on the game’s outcome than others. By using a system to estimate the ratio of high and low cards still in the deck, the technique allows players to adjust their betting and playing decisions based on the remaining composition of the deck.
Popular Card Counting Systems
Several card counting systems have been developed over the years, each with its own approach to assigning values to the cards. Here are a few notable examples:
1. Hi-Lo Count: The Hi-Lo Count is one of the simplest and most popular card counting systems. It assigns a tag of +1 to low cards (2–6), a tag of 0 to neutral cards (7–9), and a tag of −1 to high cards (10-Ace). By maintaining a running count based on these tags, players can assess the overall composition of the remaining deck.
2. KO Count: The Knock-Out (KO) Count is another popular system. In this method, all 7s, 8s, and 9s are assigned a tag of +1, while 10s through Aces are assigned a tag of −1. The remaining cards are considered neutral (tag 0).
3. Hi-Opt Systems: Hi-Opt systems, such as the Hi-Opt I and Hi-Opt II, aim to provide a more accurate assessment of the deck’s composition by considering more card values.
4. Zen Count: The Zen Count system is known for its precision in tracking the deck’s composition. It assigns a variety of values to different cards, creating a more detailed count. This system, while more complex than the other systems, can offer a greater edge to skilled players.
Additional Considerations: It’s crucial to understand that these systems vary in complexity and suitability for different players. Advanced systems like the Zen Count may offer more accuracy, but they require more practice and skill. Additionally, systems may require converting the ‘running count’ into a ‘true count’ by accounting for the number of decks remaining in the shoe. This adjustment helps in accurately determining the player’s edge.
Making Informed Decisions
By monitoring the running count and employing the chosen card counting system, players can make in-running staking decisions. When the count indicates an abundance of high cards in the remaining deck or decks, this is generally good for the player, bad for the house. In this case, players may choose to increase the size of their bets. Conversely, when the count indicates a higher proportion of low cards remaining in the deck, players may opt for smaller bets and more conservative gameplay.
Challenges and Countermeasures
Casinos are well aware of card counting strategies and have implemented various countermeasures to detect and deter such activities. They employ techniques such as automatic shuffling machines, frequent deck changes, and trained personnel to identify suspected card counters. Consequently, players who employ card counting techniques also employ camouflage methods to avoid detection. This involves blending in with other players, varying bet sizes, acting like a casual player, and avoiding suspicious behaviour.
The Evolution of Card Counting
Over the years, card counting has evolved alongside advancements in technology and changes in casino practices. The rise of online blackjack games and continuous shuffling machines (CSMs) has posed new challenges for card counters. Online casinos employ random number generators (RNGs), making it impossible to track specific cards. CSMs continuously shuffle the cards, eliminating any opportunity to gain an advantage through card counting.
Conclusion: Beating the Odds
Card counting revolutionised the game of blackjack by providing players with a mathematical strategy to gain an edge over the house. However, it requires skill and practice to implement while evading detection. Still, card counting remains a challenging yet fascinating aspect of blackjack gameplay, and players can in principle adapt their techniques to the countermeasures employed by casinos. It continues to captivate players who seek to test their skills and beat the odds at the blackjack table.
A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams, Chapman & Hall/CRC Press. 2024.
Introduction
Born in the 5th century BC in Elea (a Greek colony in southern Italy), Zeno of Elea is one of the most intriguing figures in the field of philosophy. Zeno’s paradoxes are a set of problems generally involving distance or motion. While there are many paradoxes attributed to Zeno, the most famous ones revolve around motion and are extensively discussed by Aristotle in his work, ‘Physics’. These paradoxes include the Dichotomy paradox (that motion can never start), the Achilles and the Tortoise paradox (that a faster runner can never overtake a slower one), and the Arrow paradox (that an arrow in flight is always at rest). Through these paradoxes, Zeno sought to show that our common-sense understanding of motion and change was flawed and that reality was far more complex and counterintuitive.
The Achilles and the Tortoise paradox, as one example, uses a simple footrace to question our understanding of space, time, and motion. While it’s clear in real life that a faster runner can surpass a slower one given enough time, Zeno uses the race to craft an argument where Achilles, no matter how fast he runs, can never pass a tortoise that has a head start. This thought experiment forms a remarkable philosophical argument that challenges our perceptions of reality and creates a fascinating paradox that continues to engage scholars to this day.
These paradoxes might seem simple, but they invite us into deep philosophical waters, questioning our perception of reality and illustrating the complexity of concepts we take for granted like motion, time, and distance. In this way, Zeno’s contributions continue to have profound relevance in philosophical and scientific debates, encouraging us to critically explore the world around us.
The Paradox of the Tortoise and Achilles
In one version of this paradox, a tortoise is given a 100-metre head start in a race against the Greek hero Achilles. Despite Achilles moving faster than the tortoise, the paradox argues that Achilles can never overtake the tortoise. As Aristotle recounts it, ‘In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead’.
The Underlying Infinite Process
This paradox lies in the infinite process Zeno presents. When Achilles reaches the tortoise’s original position, the tortoise has already moved a bit further. By the time Achilles reaches this new position, the tortoise has again advanced. This sequence of Achilles reaching the tortoise’s previous position and the tortoise moving further seems to continue indefinitely, suggesting an infinite process without a final, finite step. Zeno argues that this eternal chasing renders Achilles incapable of ever catching the tortoise.
A Mathematical Solution to the Paradox
The resolution to Zeno’s paradox lies in the mathematical understanding of infinite series. Using a stylised scenario where Achilles is just twice as fast as the tortoise (it’s a very quick tortoise!), we define the total distance Achilles runs (S) as an infinite series: S = 1 (the head start of the tortoise) + 1/2 (the distance the tortoise travels while Achilles covers the head start) + 1/4 + 1/8 + 1/16 + 1/32 …
By mathematical properties of geometric series, this infinite series sums to a finite value. In other words, despite there being infinitely many terms, their sum is finite: S = 2. Hence, Achilles catches the tortoise after running 200 metres, demonstrating how an infinite process can indeed have a finite conclusion.
Philosophical Implications: Is an Infinite Process Truly Resolved?
Zeno’s paradoxes, while they might be resolved mathematically, open a Pandora’s box of philosophical questions, particularly concerning the nature of infinity and the real-world interpretation of mathematical abstractions. How can a seemingly infinite process with no apparent final step culminate in a finite outcome?
The Thomson’s Lamp thought experiment, proposed by philosopher James F. Thomson, provides an insightful analogy. Imagine you have a lamp that you can switch on and off at decreasing intervals: on after one minute, off after half a minute, on after a quarter minute, and so forth, with each interval being half the duration of the previous one. Mathematically, the total time taken for this infinite sequence of events is two minutes. However, a critical philosophical question emerges at the end of the two minutes: is the lamp in the on or off state?
This question is surprisingly complex. On the one hand, you might argue that the lamp must be in some state, either on or off. However, there is no finite time at which the final switch event takes place, given the infinite sequence of switching. Hence, the state of the lamp appears indeterminate, raising questions about the applicability of infinite processes in the physical world. More prosaically, of course, you may just have blown the bulb!
This conundrum mirrors the situation in Zeno’s paradox of Achilles and the Tortoise. Just as the state of Thomson’s Lamp after the two-minute mark seems ambiguous, so does the concept of Achilles catching the tortoise after an infinite number of stages. While mathematics gives us a definitive point at which Achilles overtakes the tortoise, the philosophical interpretation of reaching this point through an infinite process is not as clear-cut.
The Thomson’s Lamp thought experiment highlights that while we can use mathematical tools to deal with infinities, interpreting these results in our finite and discrete physical world can be philosophically challenging. It reminds us that philosophy and mathematics, while often harmonious, can sometimes offer different perspectives on complex concepts like infinity, sparking ongoing debates that fuel both fields.
Zeno’s Paradoxes, the Quantum World, and Relativity
Zeno’s paradoxes, which have puzzled thinkers for millennia, find surprise echoes in the realms of quantum mechanics and the theory of relativity, two foundational components of modern physics. Thse paradoxes, originally aimed at challenging the coherence of motion and time, intersect with quantum and relativistic concepts in thought-provoking ways.
In quantum mechanics, the principle of superpoition allows particles to exist in multiple states a once until observed. This phenomenon reflects the essence of Zeno’s Arrow Paradox, where an arrow in flight is paradoxically motionless at any instant. This comparison highlights how quantum theory disrupts traditional views on motion, suggesting that at a microscopic level, movement doesn’t conform to our standard or philosophical expectations.
Meanwhile, the theory of relativity introduces the conceot of time dilation, where times appears to ‘slow down’ for an object moving at speeds close to the speed of light. This idea provides a moden perspective on Zeno’s Dichotomy Paradox, which argues that motion is impossible due to the infinite divisibility of time and space. Through relativity, we see that motion and time are relative, not absolute, concepts – illustrating a deep connection to Zeno’s philosophical challenges, even after two millennia.
Conclusion: Philosophical Debate and Contemporary Relevance
Contemporary philosophers continue to grapple with Zeno’s paradoxes, not only as historical curiosities but also as fundamental challenges to our understanding of reality. These paradoxes force us to reconsider how we conceptualise time, space, and motion. They remind us that our intuitive grasp of the world is often at odds with its underlying complexities. In today’s world, where scientific and technological advancements continually push the boundaries of what we understand, Zeno’s paradoxes remain as relevant as ever, reminding us of the enduring power and limits of human reason and the ongoing journey to comprehend the universe in which we live.
The Very Strange Implications of the Inspection Paradox
A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
The Bus Stop Scenario
Take the case of a bus that arrives, on average, every 20 minutes. It’s not a perfect rule—sometimes the bus arrives early and sometimes it’s late. But, when you calculate all the arrival times, it averages out to three times an hour, or every 20 minutes.
Now, picture yourself emerging from a side street to the bus stop, with no idea when the bus last arrived. The question that naturally arises is: how long should you expect to wait for the next bus?
Your initial thought might be, ‘Well, if it’s 20 minutes on average, then I should expect to wait around 10 minutes’. This would be halfway between the average intervals and would indeed be the case if the bus arrivals were perfectly spaced out. However, if you find yourself waiting longer than this, you might start to feel like the world is against you. The question then arises: are you just unlucky, or is something else at play?
This is where we introduce the concept of the Inspection Paradox.
Unravelling the Inspection Paradox
The Inspection Paradox is a statistical phenomenon that reveals how our expected wait times can differ from the average times we calculate, due to the randomness of our inspections or experiences.
To illustrate this, let’s look deeper into the bus scenario. The bus schedule is not as straightforward as it might seem. Remember, the bus arrives every 20 minutes on average, but not at precise 20-minute intervals. Variability changes things.
Unpredictability in the Bus Schedule
Consider a situation where half of the time, the bus arrives at an interval of 10 minutes, and the other half at an interval of 30 minutes. The overall average remains at 20 minutes, but your experience at the bus stop will differ. If you show up at the bus stop at a random time, it’s statistically more probable that you will turn up during the longer 30-minute interval than the shorter 10-minute interval.
This variation has significant implications for your expected wait time. If you land in the 30-minute interval, you can expect to wait around 15 minutes, half of that interval. If you find yourself in the 10-minute interval, you’ll only wait around 5 minutes on average. However, you’re three times more likely to hit the 30-minute gap, which means your expected wait time skews closer to 15 minutes than 5 minutes. On average, your expected wait time becomes 12.5 minutes, contrary to the intuitive answer of 10 minutes. This is calculated as follows: (3 × 15 + 1 × 5)/4 = 50/4 = 12.5 minutes.
Implications of the Inspection Paradox
This surprising realisation is the crux of the Inspection Paradox. It essentially states that when you randomly ‘inspect’ or experience an event without knowing its schedule or distribution beforehand, it often seems to take longer than the average time. This isn’t due to some cosmic force giving you a hard time; it’s simply how probability and statistics operate in the randomness of real life.
Understanding the Inspection Paradox can fundamentally change how you interpret your everyday experiences. It’s not about bad luck but rather about understanding that your perception of averages can be skewed by variability around the average.
Everyday Instances of the Inspection Paradox
Once you’re aware of the Inspection Paradox, you might start noticing it in various aspects of your everyday life.
Education Institution: Average Class Size
Consider an educational institution that reports an average class size of 30 students. Now, if you were to randomly ask students from this institution about their class size, you might find that your calculated average is higher than the reported 30.
Why does this happen?
The Inspection Paradox is at play here. If the institution has a range of small and large classes, you’re more likely to encounter students from larger classes in your random sample. This leads to a bigger average class size in your interview sample compared to the actual average class size.
Say, for example, that the institution has class sizes of either 10 or 50, and there are equal numbers of each. In this case, the overall average class size is 30. But in selecting a random student, it is five times more likely that they will come from a class of 50 students than from a class 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’. So the average class size thrown up by your survey is 5 × 50 + 1 × 10, divided by 6. This equals 260/6 = 43.3. The act of inspecting the class sizes thus increases the average obtained compared to the uninspected average. The only circumstance in which the inspected and uninspected averages coincide is when every class size is equal.
Library Study Times
Consider another scenario where you visit a library and conduct a survey asking the attendees how long they usually study. You might notice that the reported study times are generally on the higher side. This happens because the sample of students you survey is skewed towards those who spend longer times studying in the library. The reason is that the longer a student stays in the library, the higher the chance you’ll find them there during your random survey. Short-term visitors are less likely to be part of your sample, skewing the average study time upwards.
The Restaurant and the Supermarket
You might think about the implications for other scenarios, such as restaurant wait times or queue lengths at supermarkets. For the reasons we have learned about, we might expect our individual experience of waiting to be that little bit longer than a calculation of the unobserved average.
The Paradox in Other Real-Life Scenarios
Potato Digging
Why do you often accidentally cut through the biggest potato when digging in your garden? It’s because larger potatoes take up more space in the ground, increasing the likelihood of your shovel hitting them.
Downloading Files
Consider the frustration when your internet connection breaks during the download of the largest file. It’s because larger files take longer to download, increasing the window of time for potential connection issues to arise.
Conclusion: A New Lens
Understanding the Inspection Paradox equips you with a new lens through which to look at the world. It helps explain why your experiences might often differ from average expectations. It’s simply the laws of probability and statistics unfolding in a world full of randomness. With this knowledge, you can navigate the world with more informed expectations and a greater appreciation for statistical realities.
The Ship of Theseus Paradox
A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
PLUTARCH’S PARADOX
The Ship of Theseus Paradox has its roots in ancient Greek philosophy, emerging as a crucial discussion point in understanding identity and change. Originally posed by the philosopher Plutarch, the paradox was used to question whether a ship, which was gradually having all its wooden parts replaced, remained fundamentally the same ship. This paradox was not just a mere intellectual exercise; it was deeply rooted in the Greek exploration of ‘being’ and ‘becoming’, which were crucial themes in their philosophical inquiries. Over time, the Ship of Theseus became a pivotal reference in philosophical discussions about identity, persisting through the centuries as a tool to test the limits of our understanding of continuity and change.
A QUESTION OF IDENTITY
The Ship of Theseus Paradox is central to discussions in philosophy regarding the nature of identity. It presents a compelling challenge to the idea of persistent identity over time, particularly when an object undergoes gradual change.
THE LEGEND
The story of Theseus’s Ship begins with the legendary hero Theseus, who sailed on a ship to the island of Crete to defeat the Minotaur. After his victory, his ship was preserved and displayed in Athens as a symbol of the city’s pride. Over time, the wooden planks of the ship began to decay and were replaced with new ones. Eventually, every original piece of the ship was replaced, leading to the question: Is the ship still the same ship that Theseus sailed on, even though none of its original components remain?
CONTINUITY AND IDENTITY
If an object has all its parts replaced, is it still the same object? If we say that it is the same object, then we must explain why and how it retains its identity despite having none of its original components. Conversely, if we say that it is not the same object, then we must determine at what point it ceased to be the original and became something new.
The question of whether an object remains the same when its parts have been entirely replaced makes us reassess our understanding of what constitutes an object’s identity. Are objects defined by the matter of which they’re composed, their structure, their history, or by a combination of these and maybe other factors?
THE SUBSTANCE VIEW
The Substance View proposes that the identity of an object is tied to the substance or the matter it is made of. According to this perspective, the Ship of Theseus depends on the continuity of the material components that constitute it. When all the original parts of the ship are replaced, the ship loses its original identity and becomes a new object. This view sees the ship’s identity as static, fixed, and dependent on its material constituents.
This interpretation faces challenges when considering gradual transformations, as it becomes difficult to pinpoint the exact moment when the ship’s identity changes. Moreover, this view might struggle to account for the importance of functional and relational aspect of objects. Critics argue that it cannot satisfactorily explain cases where an object’s function and relation to the world remain constant despite material changes.
Recent debates have also brought into question the implications of digital and virtual identities. In a digital era, where replication and modification of virtual entities are commonplace, how does the Ship of Theseus Paradox inform our understanding of digital identity? Does a digital object lose its ‘identity’ when its code is altered or does it transcend traditional notions of materiality?
THE RELATIONAL VIEW
The Relational View focuses on the idea that the identity of an object is grounded in its relationships with other objects and entities.
Supporters of the Relational View argue that the Ship of Theseus retains its identity through its connections to the story of Theseus, its role in the society in which it exists, and the memories and associations that people have with it.
THE BUNDLE THEORY
The Bundle Theory suggests that an object is nothing more than a bundle of its properties—there’s no ‘object’ beyond the collection of its characteristics. Applying this theory to the Ship of Theseus, one might argue that the ship is merely a bundle of its properties such as its shape, size, purpose, and the arrangement of its planks. As these properties change (when the planks are replaced), the ship’s identity changes too. However, if the ship retains its structure, function, and perhaps other properties, it can still be recognised as the ‘same’ ship. This interpretation encourages us to think of objects as collections of properties rather than stable, unchanging entities.
ARTEFACTS
In the context of the Ship of Theseus Paradox and the discussion on identity and change, the restoration of historical artefacts offers a compelling parallel.
Restoration and Identity
The process of restoring historical artefacts often involves repairing or replacing deteriorated components with new materials to preserve the artefact’s appearance, function, or structural integrity. This process raises questions similar to those in the Ship of Theseus: does an artefact maintain its original identity after restoration, especially when significant portions have been replaced or altered?
Authenticity vs. Preservation
The challenge in artefact restoration lies in balancing authenticity with preservation. Authenticity refers to the degree to which an artefact remains unchanged, retaining its original materials and form. On the other hand, preservation might require the introduction of new materials to prevent further decay or to restore an artefact to a former state. At what point does an artefact become a replica rather than an original?
CASE STUDIES
The Sistine Chapel
Consider the restoration of the Sistine Chapel ceiling, where layers of grime and soot were removed to reveal Michelangelo’s original colours. Some critics argued that the vibrant colours revealed by the restoration were inconsistent with Michelangelo’s intentions, suggesting that the restoration had altered the fresco’s identity. Others contended that the restoration brought the artwork closer to its original state, thus preserving its true identity.
The Parthenon
Similarly, the restoration of ancient buildings, like the Parthenon in Athens, involves replacing eroded stones with new material. Critics might question whether the building maintains its original identity after such changes, while proponents argue that restoration helps preserve the structure’s historical and cultural significance. A key issue is whether it is more ‘genuine’ as a ruin bearing the marks of its history or restored to a state believed to be true to its original form.
The Last Supper
“The Last Supper” by Leonardo da Vinci has undergone several restorations over the centuries due to deterioration caused by environmental factors, wartime damage, and previous restoration attempts. Each restoration has presented a dilemma, requiring restorers to decide whether to attempt to revert the mural to its original state (as much as possible) or to stabilise its condition to prevent further degradation.
Critics argue that each layer of restoration moves the painting further from Leonardo’s original vision, potentially altering its identity. They contend that the original materials, brushstrokes, and techniques employed by Da Vinci contribute fundamentally to the painting’s essence and that replacing or significantly altering these elements diminishes the work’s authenticity.
In philosophical terms, the restoration of “The Last Supper” mirrors the Ship of Theseus Paradox by raising questions about continuity and identity over time. If all the original pigment is removed and replaced, is it still the same painting? Or does the essence of the artwork lie in its visual appearance, its historical significance, or the intent behind its creation?
The Bridge at Mostar
Originally built in the 16th century by the Ottomans, the Stari Most stood as a symbol of unity and an architectural marvel, connecting the diverse communities in Mostar across the Neretva River. Its wartime destruction in 1993 became a poignant symbol of cultural and communal fragmentation,
The decision to rebuild the Stari Most was fraught with questions about identity and authenticity. Could a reconstructed bridge, built centuries after the original, serve the same symbolic and functional roles as its predecessor? The reconstruction effort aimed to use original techniques and materials as much as possible, sourcing local stone and employing traditional Ottoman construction methods. This approach sought to preserve the bridge’s historical authenticity and cultural significance, even as it acknowledged the impossibility of an exact physical replica.
The Stari Most’s reconstruction challenges the Ship of Theseus Paradox by asking whether an object—destroyed and subsequently rebuilt with the intent of mirroring the original as closely as possible—retains its identity. This case pushes the paradox further by introducing the element of complete destruction rather than gradual replacement. Is the new bridge the same as the old, despite the interruption of its physical existence? Or does its reconstruction, imbued with the collective memory, effort, and intention to bridge past and present, confer upon it a renewed identity that is both continuous and distinct?
Through its destruction and reconstruction, the Stari Most offers a powerful narrative on the complexities of identity, continuity, and change. It exemplifies how reconstructed heritage can carry forward the essence of the original, serving as a bridge not only in physical space but in time, memory, and meaning, thereby engaging with the philosophical inquiries posed by the Ship of Theseus Paradox in a deeply human context.
VIRTUAL IDENTITIES
Recent debates have also brought into question the implications of digital and virtual identities. In a digital era, where replication and modification of virtual entities are commonplace, how does the Ship of Theseus Paradox inform our understanding of digital identity? Does a digital object lose its ‘identity’ when its code is altered or does it transcend traditional notions of materiality?
Digital Personas
Digital personas are curated representations of ourselves on the internet, shaped by the information we choose to share on social media, forums, and other online platforms. These personas are not static; they evolve as we update our profiles, post new content, and interact with others. This fluidity raises questions akin to those posed by the Ship of Theseus: if a digital persona is constantly changing, at what point does it become fundamentally different from its original incarnation? Moreover, the curated nature of digital personas prompts us to consider which aspects of our identity are essential and which are mutable.
Artificial Intelligence
AI presents a more complex challenge to traditional concepts of identity. Machine learning algorithms allow AI systems to evolve based on new data and experiences, much like humans learn and change over time. This adaptability leads to questions about the continuity of identity: if an AI’s decision-making processes and behaviours change significantly, is it still the ‘same’ AI?
IMPLICATIONS FOR PERSONAL IDENTITY
In terms of personal identity, the Ship of Theseus Paradox intersects significantly with theories of psychological continuity. According to this theory, personal identity is maintained through the continuity of psychological features like memory, personality, and consciousness. If we apply this to the Ship of Theseus, it raises the question: Is identity maintained through physical continuity or through the continuity of function and recognition?
This perspective is particularly relevant in discussions about human development and change. As individuals undergo physical, emotional, and psychological changes throughout life, at what point do they become ‘different’ individuals, if at all? The Ship of Theseus Paradox, thus, serves as a metaphor for exploring the fluidity and resilience of personal identity amidst constant change.
By integrating these aspects, the discussion around the Ship of Theseus Paradox becomes not only more historically grounded and analytically rich but also deeply connected to contemporary and personal contexts.
BROADER IMPLICATIONS
The Ship of Theseus Paradox may provide a useful framework for grappling with emerging ethical and philosophical issues as advancements in technology push the boundaries of what is possible. For instance, questions about the continuity of consciousness and the identity of entities that undergo substantial change arise in fields such as artificial intelligence and human augmentation.
CONCLUSION: CHALLENGING OUR ASSUMPTIONS
The Ship of Theseus Paradox remains an engaging and relevant tool for philosophical inquiry. Its exploration of identity and change continues to resonate with modern audiences. By challenging our assumptions and forcing us to question our understanding of th world, the paradox fulfils a key purpose of any paradox: to provoke thought and inspire exploration.
A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
The Conviction
In the final weeks of the 20th century, a lawyer named Sally Clark was convicted of the murder of her two infant sons. Despite being a woman of good standing with no history of violent behaviour, Clark was swept up in a whirlwind of accusations, trials, and appeals that would besmirch the criminal justice system and cost her dearly.
The Investigation and Trial—Building a Case on Uncertainty
The deaths of Clark’s two children were initially assumed to be tragic instances of Sudden Infant Death Syndrome (SIDS), a cause of infant mortality that was not well understood even by medical experts. However, the authorities became suspicious of the coincidental deaths, leading to Clark’s eventual trial. As the investigation evolved, it subsequently transpired that numerous pieces of evidence helpful to the defence were withheld from them.
Statistical Evidence—The Misinterpretation
The prosecution presented a piece of seemingly damning statistical evidence during Clark’s trial. One of their witnesses, a paediatrician, asserted that the probability of two infants from the same family dying from SIDS was incredibly low—approximately 1 in 73 million. He compared the odds to winning a bet on a longshot in the iconic Grand National horse race four years in a row.
The Prosecutor’s Fallacy—The Dangerous Conflation of Probabilities
The flaws in the statistical argument presented at the trial were both substantial and consequential. The paediatrician had mistakenly assumed that the deaths of Clark’s children were unrelated, or ‘independent’ events. This assumption neglects the potential for an underlying familial or genetic factor that might contribute to SIDS.
Moreover, the paediatrician’s argument represents a common misinterpretation of probability known as the ‘Prosecutor’s Fallacy’. This fallacy involves conflating the probability of observing specific evidence if a hypothesis is true, with the probability that the hypothesis is true given that evidence. These are two very different things but easy for a jury of laymen to confuse.
The Prosecutor’s Fallacy Explained
This fallacy arises from confusing two different probabilities:
1. The probability of observing specific evidence (in this case, two SIDS deaths) if a hypothesis (Clark’s guilt) is true.
2. The probability that the hypothesis is true given the observed evidence.
The Need for Comparative Likelihood Assessment
The Royal Statistical Society emphasised the need to compare the likelihood of the deaths under each hypothesis—SIDS or murder. The rarity of two SIDS deaths alone doesn’t provide sufficient grounds for a murder conviction.
Prior Probability—Understanding the Likelihood of Guilt before the Evidence
Prior probability—a concept integral to understanding the Prosecutor’s Fallacy, and fundamental to Bayesian reasoning, is often overlooked in court proceedings. This term refers to the probability of a hypothesis (in this case, that Sally Clark is a child killer) being true before any evidence is presented.
Given that she had no history of violence or harm towards her children, or anyone else, or any indication of such a tendency, the prior probability of her being a murderer would be extremely low. In fact, the occurrence of two cases of SIDS in a single family is much more common than a mother murdering her two children.
The jury should weigh up the relative likelihood of the two competing explanations for the deaths. Which is more likely? Double infant murder by a mother or double SIDS?
More generally, it is likely in any large enough population that one or more cases of something highly improbable will occur in any particular case.
In a letter from the President of the Royal Statistical Society to the Lord Chancellor, Professor Peter Green explained the issue succinctly:
The jury needs to weigh up two competing explanations for the babies’ deaths: SIDS or murder. The fact that two deaths by SIDS is quite unlikely is, taken alone, of little value. Two deaths by murder may well be even more unlikely. What matters is the relative likelihood of the deaths under each explanation, not just how unlikely they are under one explanation.
Put another way, before considering the evidence, the prior probability of Clark being a murderer, given her background and lack of violent history, was extremely low. The probability of two SIDS deaths in one family, while rare, was still significantly higher than the likelihood of a mother murdering her two children.
The Need for Comparative Likelihood Assessment
The Royal Statistical Society emphasised the need to compare the likelihood of the deaths under each hypothesis—SIDS or murder. The rarity of two SIDS deaths alone doesn’t provide sufficient grounds for a murder conviction. More recently, the Royal Statistical Society has weighed in on the strange and troubling case in the UK of Lucy Letby, convicted on disputed circumstantial evidence. So far, the justice system seems to have been paying no attention. Will that change? We shall see – but on past experience it might seem rather unlikely.
The Case of Lottie Jones
To illustrate the Prosecutor’s Fallacy, consider the fictional case of Lottie Jones, charged with winning the lottery by cheating. The fallacy occurs when the expert witness equates the low probability of winning the lottery (1 in 45 million) with the probability that a lottery win was achieved unfairly.
As in the Sally Clark case, the prosecution witness in this fictional parody commits the classic ‘Prosecutor’s Fallacy’. He assumes that the probability Lottie is innocent of cheating, given that she won the Lottery, is the same thing as the probability of her winning the Lottery if she is innocent of cheating. The former probability is astronomically higher than the latter unless we have some other indication that Lottie has cheated to win the Lottery. It is a clear example of how it is likely, in any large enough population, that things will happen that are improbable in any particular case. In other words, the 1 in 45 million represents the probability that a Lottery entry at random will win the jackpot, not the probability that a player who has won did so fairly!
Lottie just got very, very lucky just as Sally Clark got very, very unlucky.
The Aftermath—Tragedy and Lessons Learned
Following her acquittal in 2003, Sally Clark never recovered from her ordeal and sadly died just a few years later. Her story stands as testament to the potential for disastrous consequences when statistics are misunderstood or misrepresented. Even when acquitting her, the judgment was based primarily on other evidence, side-lining the role that the proper application of statistics and Bayesian probability should have brought to the case.
O.J. Simpson—An Alternate Scenario
Even in high-profile cases, such as American former actor and NFL football star O.J. Simpson’s murder trial in the 1990s, this same misinterpretation of statistics is prevalent. Simpson’s defence team argued that it was unlikely Simpson killed his wife because only a small percentage of spousal abuse cases result in the spouse’s death. This argument, though statistically accurate, overlooks the relevant information—the fact that about 1 in 3 murdered women were killed by a spouse or partner. This represents a very clear case of the misuse of the Inverse or Prosecutor’s Fallacy in argumentation before a jury.
Conclusion: The Importance of Statistical Literacy
The importance of statistics in our justice system cannot be overstated. We must recognise the potential for misinterpretation and the potentially devastating results. A concerted effort to promote statistical literacy, particularly within our legal systems, can, if heeded, go a long way in preventing future miscarriages of justice, and rectifying current ones. In truth, however, very little if any progress has been made in this regard, and we have a very, very long way to go!
A version of this article appears in TWISTED LOGIC: Paradoxes, Puzzles, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC. 2024.
Size Matters
What is the minimum number of individuals that need to be present in the room for it to be more likely than not that at least two of them share a birthday? This is what the ‘Birthday Problem’ seeks to solve.
For the sake of simplicity, let’s assume that all calendar dates have an equal chance of being someone’s birthday and let’s disregard the Leap Year occurrence of 29 February.
A Basic Intuition: Analysing the Odds
At first glance, you might think that the odds of two people sharing a birthday are incredibly low. In a group of just two people, the likelihood of them sharing a birthday is a mere 1/365. Why is that? We have 365 days in a year, hence there’s only one chance in 365 that the second person would have been born on the same specific day as the first person.
Now, let’s take a group of 366 people. In this case, it’s certain that at least one person shares a birthday with someone else, due to the simple fact that we only have 365 possible birthdays (ignoring Leap Years).
The initial intuition may suggest that the tipping point—the group size at which there’s a 50% chance of two individuals sharing a birthday—is around the midpoint of these two extremes. You may think it lies around a group size of 180. However, the reality is surprisingly different, and the actual answer is much smaller.
The Calculations: Unravelling the Birthday Paradox
To understand the concept better, we need to dig deeper into the probabilities involved. Let’s consider a duo: Julia and Julian. Let’s assume that Julia’s birthday falls on 1 May. The chance that Julian shares the same birthday, assuming an equal distribution of birthdays across the year, is 1/365.
What about the probability that Julian doesn’t share a birthday with Julia? It’s simply 1 minus 1/365, or 364/365. This number illustrates the chance that the second person in a random duo has a different birthday than the first person.
Adding a third person into the mix changes things slightly. The chance that all three birthdays are different is the chance that the first two are different (364/365) multiplied by the probability that the third birthday is unique (363/365). So, the probability of three different birthdays equals (364/365) × (363/365).
As we expand the group, the calculations continue in a similar manner. The more people in the room, the higher the chances of finding at least two people sharing a birthday.
Consider a group of four people. The probability that four people have different birthdays is (364 × 363 × 362)/(365 × 365 × 365). To find the probability that at least two of the four share a birthday, we subtract this number from 1. Thus, the odds of having at least two people with the same birthday in a group of four are about 1.6%.
As the number of people in the room increases, the probability of at least two sharing a birthday grows:
• With 5 people, the probability is 2.7%.
• With 10 people, the probability is 11.7%.
• With 16 people, the probability is 28.1%.
• With 23 people, the probability is 50.5%.
• With 32 people, the probability is 75.4%.
• With 40 people, the probability is 89.2%.
The Paradox Unveiled: It’s Not Just about Birthdays
You might be wondering why we need just 23 people to reach a 50% chance of finding shared birthdays. This can be explained by how many possible pairs can be made in a group. In a group of 23, there are 253 unique pairs. Each of these pairs has a 1/365 chance of sharing a birthday, and all these possibilities add up. This is what makes the birthday problem so counterintuitive. Basically, when a large group is analysed, there are so many potential pairings that it becomes statistically likely for coincidental matches to occur.
This is a perfect demonstration of the concept of multiple comparisons and an example of the so-called ‘Multiple Comparisons Fallacy’.
The same reasoning applies to balls being randomly dropped into open boxes. Assume there is an equal chance that a ball will drop into any of the individual boxes. If there are 365 such boxes, into which 23 balls are randomly dropped, with an equal chance that a ball will drop into any specific box. Now, there is just over a 50% chance that there will be at least two balls in at least one of the boxes. Randomness produces more aggregation than intuition leads us to expect.
Your Personal Birthday Chances: Where Do You Stand?
The reason for the paradox, therefore, is that the question is not asking about the chance that someone shares your particular birthday or any particular birthday. It is asking whether any two people share any birthday.
While the birthday problem shows the increased likelihood of shared birthdays in a group, the chance that someone shares your birthday specifically is a different question.
In a group of 23 people, including yourself, the probability that at least one person shares your birthday is much lower than 50%—it’s about 6%. This is because there are only 22 potential pairings that include you.
Even in a group of 366 people, the probability that someone shares your specific birthday is only around 63%.
Conclusion: The Magic of Probability and the Birthday Paradox
The Birthday Paradox reveals an intriguing counterintuitive fact about probability: a group of just 23 people has a greater than 50% chance of including at least two people who share the same birthday. It sheds light on the intricacies of probability by demonstrating how many opportunities there are for matches to occur, even in seemingly small groups. For example, if you can find out the birthdays of the 22 players at the start of a football game, and the referee, more than half of the time two of them will share a birthday.
This fascinating concept has applications way beyond birthdays. It’s also very important for the safety and performance of computer systems and online security. This idea helps specialists prevent and deal with issues that occur when data unexpectedly overlaps. Understanding the paradox is crucial, therefore, for those who design and secure computer systems, helping them to make these systems more reliable and efficient.
Nevertheless, it’s in the social setting of parties where the paradox becomes a delightful surprise. Next time you’re among friends or at any casual meet-up, consider introducing this paradox; you might just bring to life the unexpected magic of probability!
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
The Existential Coin Toss is a thought experiment where the existence of two different worlds depends on a coin flip. World A (Heads) has one black-bearded individual, while World B (Tails) has two individuals, one with a black and another with a brown beard. Waking up in one of these worlds without prior knowledge of your world or beard colour, what probability would you assign to being in World B?
The Set-Up
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Heads creates World A, which is inhabited by one black-bearded individual.
Tails brings forth World B, which is populated by two individuals, one with a black beard and the other with a brown beard.
Awakening in the darkness, unaware of your world and beard colour, but aware of the rules of your existence, what probability would you assign to the coin having landed on Tails, placing you in World B?
The answer hinges on your basic assumptions about existence.
The Self-Sampling Assumption (SSA)
This approach encourages us to think of ourselves as a random selection from all entities that could have been us—our ‘reference class’. Consequently, we are a randomly selected bearded individual, with an equal chance of living in World A (Heads) or World B (Tails). If in World B, there’s a 50-50 probability of sporting either a black or brown beard.
But what happens when the light comes on and you see a black beard? Now, the probability of being in World A, where the sole inhabitant has a black beard, increases. Given the choice between World A (100% black beard chance) and World B (50% black beard chance), the likelihood of residing in World A is twice as much, making it a 2/3 chance the coin landed Heads.
The Self-Indication Assumption (SIA)
This alternative perspective suggests that you are twice as likely to be in a world where two observers exist than in a world with just one observer. Thus, you might lean towards World B, in which there are two observers, giving it twice the likelihood (2/3 chance) as World A (1/3 chance). But, once the lights are on and your beard is revealed to be black, the probability of living in World B reduces to 1/2, the same as the probability of living in World A, since your existence is confirmed in a scenario where both worlds have equal chances of your specific condition.
Implications
The contrasting perspectives of SSA and SIA in the Existential Coin toss thought experiment illustrate the complexities in assigning probabilities to our own existence. The assumptions we choose significantly influence our conclusions about our likelihood of existing in one world rather than another. As such, this not only sheds light on philosophical debates surrounding conditional probability but also challenges our understanding of existence and identity in uncertain contexts.
Unravelling the Sleeping Beauty Problem
The Sleeping Beauty Problem puts the Self-Sampling Assumption (SSA) and Self-Indication Assumption (SIA) to the test.
In this Problem, Sleeping Beauty volunteers for an experiment where she goes to sleep. A fair coin will be tossed to determine the next steps:
• If the coin lands on Heads, she will be awakened once (on Monday).
• If it lands on Tails, she will be awakened twice (on Monday and Tuesday).
On both awakenings, she has no memory of any previous awakenings, and thus can’t tell which day it is or how many times she’s been awoken. When she wakes, she’s asked: ‘What chance do you assign to the proposition that the coin landed Heads?’
Adopting the Self-Sampling Assumption (SSA)
Under this assumption, Sleeping Beauty would argue that there’s a 50-50 chance the coin landed on Heads or Tails, as those are the only two possible outcomes from a fair coin toss. This perspective doesn’t change upon waking. Only if she’s told that it’s her second awakening (which means it’s Tuesday and the coin must have landed Tails) will she change her belief to 100% Tails and 0% Heads.
Adopting the Self-Indication Assumption (SIA)
Under this assumption, Sleeping Beauty considers the number of observer-moments—points at which she is awake and observing. There are two such points if the coin lands Tails (one on Monday and one on Tuesday), but only one if the coin lands Heads (on Monday).
From this viewpoint, Sleeping Beauty would reason there’s a 1/3 chance the coin landed Heads and a 2/3 chance it landed Tails. This is because there are three observer-moments in total (Monday on Heads, Monday on Tails, and Tuesday on Tails), and each one is equally likely. So the coin landing Tails (which creates two observer-moments) is twice as likely as the coin having landed Heads (which creates only one observer-moment).
In summary, the Sleeping Beauty Problem involves SSA and SIA to determine probabilities based on the number of awakenings. Here, the SSA leads to a 50% chance of heads, while the SIA suggests a 1/3 probability, due to more observer-moments under tails.
Thus, the SSA and SIA lead to different conclusions in the Sleeping Beauty Problem, just as in the God’s Coin Toss problem. The correct approach remains a topic of debate among philosophers and statisticians, reflecting broader inquiries into how we interpret probability and make decisions under uncertainty.
Dilemma of the Presumptuous Philosopher
The Presumptuous Philosopher Problem introduces a critical examination of the Self-Indication Assumption (SIA) by presenting a scenario where SIA seems to lead to counterintuitive or problematic conclusions.
Consider a situation where scientists are evaluating two theories, each equally supported by prior evidence. Theory 1 predicts a universe with a million times more observers than Theory 2, but new evidence from a particle accelerator now strongly supports Theory 2. Despite the empirical evidence supporting Theory 2, philosophers using the SIA can insist that Theory 1 is much more likely to be correct. Look at it this way. You exist, and Theory 1 makes your existence a million times more likely than Theory 2, because there are a million times more observers that exist if Theory 1 is true than if Theory 2 is true. To put it another way, given the fact that you exist, a case can be made for supporting a theory that proposes that a very large number of observers exist over a theory that proposes a much smaller number, even if empirical evidence strongly contradicts it.
But should the sheer number of potential observers really sway our belief in a theory, especially when faced with concrete evidence to the contrary? Critics argue that this perspective cam lead to presumptuous conclusions, hence the name of the problem. In this way, the Presumptuous Philosopher Problem highlights the tension beytwee this approach and traditional evidence-based reasoning.
Conclusion: Choosing Our Assumptions
These thought experiments illustrate the complexity of probability and existence. They challenge us to ponder: Which assumption aligns with our intuition, and how reliable is our intuition in such abstract scenarios?
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
A Dive into the Two Envelopes Paradox: Unravelling the Enigma
The ‘Two Envelopes Paradox’, also known as the ‘Exchange Paradox’, is a classic conundrum of choice and value. This deceptively simple dilemma presents us with two envelopes, each containing a certain amount of money. The rules of the dilemma are simple. One envelope contains exactly twice as much as the other. We choose an envelope, look inside, and then face the option of sticking with our original choice or switching to the other envelope.
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The Allure of the Switch
At first glance, the decision seems straightforward. We stand to double our money by switching. Let’s say the chosen envelope contains £100. In that case, the other envelope either contains £50 (half) or £200 (double). It’s tempting to switch, as we could either gain £100 or lose £50. It appears that on average we are better off switching, no matter what amount lies in the initial envelope, since we are equally likely to gain £100 as to lose £50.
The allure of the switch persists even when we do not know the envelope’s contents. We could argue that if the chosen envelope holds X pounds, then the amount in the other envelope would be either 2X or 1/2X, with equal likelihood. Mathematically, this can be shown to equate to an expected value of ½ (2½X), or 5/4X, which is greater than X. On this basis, it seems a good idea to switch envelopes.
An Infinite Dilemma: The Absurdity of the Switch
Following this line of logic might lead us to a bewildering conclusion. Why not switch back and forth between the envelopes endlessly? If each switch supposedly increases the expected value, would we not become ever wealthier by just continually swapping envelopes?
This conclusion defies our sense of reality. We know that there’s something fundamentally wrong with the idea of a perpetual money-making machine created just by swapping envelopes. Yet, where does our logic fail us?
Stepping Back: Viewing the Total Picture
A different approach to the problem is to consider the total sum of money present in both envelopes. Let’s represent this total as A. Since one envelope contains Y pounds and the other has twice as much, 2Y, we know that A equals 3Y.
If we initially picked the envelope with Y, switching to the 2Y envelope would give us an additional Y. However, if our first choice was the 2Y envelope, switching to the Y envelope would result in the loss of Y. So, there is an equal chance of gaining Y as it does of losing Y by making the switch. Balancing out these probabilities, we can conclude that the expected gain from switching is precisely zero.
Resolving the Paradox: Framing the Problem Correctly
A key reason why the paradox seems so puzzling lies in how we frame the situation. The argument for switching implies that there are three possible amounts of money in play: X, 2X, or 1/2X. However, we know that there are only two envelopes, hence only two possible amounts.
By accurately framing the problem with just two amounts of money, we realise there is no expected gain or loss from switching envelopes. This remains true whether we frame the amounts as X and 2X or as X and 1/2X. Regardless, the average gain from switching equals zero.
Conclusion: Embracing the Mystery of Probability
The Two Envelopes Paradox demonstrates the often counterintuitive nature of probability and expected value. Despite the tempting initial logic suggesting a continual switch might be profitable, careful consideration reveals that there is, in fact, no inherent benefit to switching—a twist that showcases the often-mystifying appeal of mathematical reasoning.
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
A Thought Experiment
The Sleeping Beauty Problem is a thought experiment that challenges our understanding of probability. It involves Sleeping Beauty, a coin toss, and a scenario where her memory is erased, leading to a debate between two main schools of thought: ‘halfers’ and ‘thirders’.
The Sleeping Beauty Experiment
The experiment plays out as follows: Sleeping Beauty participates in an experiment, starting on a Sunday. The course of the experiment depends entirely on the outcome of a fair coin toss. If it lands heads, Beauty will be woken and interviewed only on Monday. If it lands tails, she will be awakened and interviewed on both Monday and Tuesday. On each occasion, she is asked what chance she assigns to the coin having landed heads. After she answers, she is put back into a sleep with a drug that erases her memory of that awakening. The experiment in any case finishes on Wednesday, with Sleeping Beauty waking up without an interview.
In other words, Sleeping Beauty participates in a coin toss experiment. If the coin lands heads, she is woken and interviewed only on Monday. If tails, she is woken on both Monday and Tuesday, with each awakening followed by memory erasure. She is asked each time about the likelihood of the coin landing heads.
Probability Paradox: Halfers vs. Thirders
When presented with this experiment, two primary interpretations of how Sleeping Beauty should calculate the probability emerge. Halfers propose that since the coin is only tossed once and no new information is collected by Beauty, she should assert a 1 in 2 chance that the coin landed heads. On the contrary, Thirders argue that from Sleeping Beauty’s standpoint, there are three equally likely scenarios, two of which involve the coin landing tails and one after a heads. Specifically, these are:
- It landed heads, and it is Monday.
- It landed tails, and it is Monday.
- It landed tails, and it is Tuesday.
Therefore, Thirders suggest that whenever she wakes up, she should assign a 1 in 3 chance to the coin having landed heads.
The Betting Frame: Determining Fair Odds in the Sleeping Beauty Problem
One potential strategy for deciphering this complex issue is by considering it in terms of fair betting odds. For instance, if Sleeping Beauty were offered odds of 2 to 1 (£1 to win a net £2) that the coin landed heads, should she take the bet?
The best way to look at this is to think about what would happen if she accepted the 2 to 1 odds each time she woke up. If the coin toss results in heads, she’d be woken up once, bet £10, and profit £20. But if the coin lands on tails, she’d be woken up twice, place two £10 bets (a total of £20) and lose both times.
Her ‘average’ result with this betting strategy would be to break even. This implies that 2 to 1 represent the correct odds. These odds (£1 win to win a net £2) are consistent with a probability of 1/3. So, using this betting test, when Beauty wakes up, she should think there’s a 1 in 3 chance that the coin landed on heads. This supports the ‘Thirder’ case.
Shifting Probabilities: From Unconditional to Conditional
A critical step in unravelling this puzzle involves an examination of the ‘prior probability’. This is the probability assigned before the collection of any new information. If asked to estimate the likelihood of a fair coin landing heads without any additional conditions, Beauty should answer 1/2. However, with added information, the question can be reformulated into estimating the probability of her waking as a result of the coin landing heads. Here, thirders would argue for a 1/3 probability. So, what information does Beauty actually have when she goes to sleep that Sunday, and how does that affect the prior probability that she should assign to the coin landing heads? Bear in mind, though, that the coin is only tossed once, and it is a fair coin.
Conclusion: How the Sleeping Beauty Problem Combines Chance and Deep Thought
The Sleeping Beauty Problem is more than a statistical puzzle; it’s a probe into the nature of information and observation. It shows that our understanding of probability can significantly shift based on the framing of the question and the information available to us. Indeed, it shakes up how we think and makes us wonder about what ‘information’ really is. This serves as a powerful reminder that the real world, like the Sleeping Beauty Problem, doesn’t always have easy or clear-cut answers. The more we dig into this mind-bending problem, the more we learn from it.
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
Card Counting: A Winning Strategy in Blackjack
In 1962, Ed Thorp introduced a strategy that would forever change the landscape of blackjack: card counting. His book, Beat the Dealer: A Winning Strategy for the Game of Twenty-One, presented a system based on probability theory that allowed players to gain an advantage over the house. Since then, card counting has become a topic of fascination for blackjack players worldwide.
Understanding the Basics of Blackjack
To grasp the significance of card counting, it’s essential to understand the fundamentals of blackjack. The basic objective of the game is simple: players aim to draw cards that beat the dealer’s hand without exceeding a total of 21. While basic strategy provides players with a foundation for optimal gameplay, card counting takes it a step further by incorporating the knowledge of which cards have already been dealt.
The Concept of Card Counting
Card counting revolves around the concept that certain cards have a different impact on the game’s outcome than others. By using a system to estimate the ratio of high and low cards still in the deck, the technique allows players to adjust their betting and playing decisions based on the remaining composition of the deck.
Popular Card Counting Systems
Several card counting systems have been developed over the years, each with its own approach to assigning values to the cards. Here are a few notable examples:
- Hi-Lo Count: The Hi-Lo Count is one of the simplest and most popular card counting systems. It assigns a tag of +1 to low cards (2–6), a tag of 0 to neutral cards (7–9), and a tag of −1 to high cards (10-Ace). By maintaining a running count based on these tags, players can assess the overall composition of the remaining deck.
- KO Count: The Knock-Out (KO) Count is another popular system. In this method, all 7s, 8s, and 9s are assigned a tag of +1, while 10s through Aces are assigned a tag of −1. The remaining cards are considered neutral (tag 0).
- Hi-Opt Systems: Hi-Opt systems, such as the Hi-Opt I and Hi-Opt II, aim to provide a more accurate assessment of the deck’s composition by considering more card values.
- Zen Count: The Zen Count system is known for its precision in tracking the deck’s composition. It assigns a variety of values to different cards, creating a more detailed count. This system, while more complex than the other systems, can offer a greater edge to skilled players.
Additional Considerations: It’s crucial to understand that these systems vary in complexity and suitability for different players. Advanced systems like the Zen Count may offer more accuracy, but they require more practice and skill. Additionally, systems may require converting the ‘running count’ into a ‘true count’ by accounting for the number of decks remaining in the shoe. This adjustment helps in accurately determining the player’s edge.
Making Informed Decisions
By monitoring the running count and employing the chosen card counting system, players can make in-running staking decisions. When the count indicates an abundance of high cards in the remaining deck or decks, this is generally good for the player, bad for the house. In this case, players may choose to increase the size of their bets. Conversely, when the count indicates a higher proportion of low cards remaining in the deck, players may opt for smaller bets and more conservative gameplay.
Challenges and Countermeasures
Casinos are well aware of card counting strategies and have implemented various countermeasures to detect and deter such activities. They employ techniques such as automatic shuffling machines, frequent deck changes, and trained personnel to identify suspected card counters. Consequently, players who employ card counting techniques also employ camouflage methods to avoid detection. This involves blending in with other players, varying bet sizes, acting like a casual player, and avoiding suspicious behaviour.
The Evolution of Card Counting
Over the years, card counting has evolved alongside advancements in technology and changes in casino practices. The rise of online blackjack games and continuous shuffling machines (CSMs) has posed new challenges for card counters. Online casinos employ random number generators (RNGs), making it impossible to track specific cards. CSMs continuously shuffle the cards, eliminating any opportunity to gain an advantage through card counting.
Conclusion: Beating the Odds
Card counting revolutionised the game of blackjack by providing players with a mathematical strategy to gain an edge over the house. However, it requires skill and practice to implement while evading detection. Still, card counting remains a challenging yet fascinating aspect of blackjack gameplay, and players can in principle adapt their techniques to the countermeasures employed by casinos. It continues to captivate players who seek to test their skills and beat the odds at the blackjack table.
Professor Leighton Vaughan Williams 