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Happy New Year!!

New Year’s Article – 2025!

Champagne, Lotteries and the Science of Probability

by Ian Sherrington
December 2024

Series – The Online Gambling Industry
The Paperwork – Jurisdictions, Legal and Compliance

As the clock strikes midnight on New Year’s Eve, the world is united in celebration. Glasses of champagne fizz, hopeful punters clutch lottery tickets, and the air buzzes with anticipation of a brand new year.

At the heart of these New Year’s customs lies the fascinating science of probability – the core of the gambling industry.

Probability is the mathematical framework that allows us to quantify uncertainty, to make sense of randomness, and to understand the odds of seemingly unpredictable events. It’s what makes the journey of champagne bubbles captivating and the dream of a lottery jackpot tantalizing. And it’s a field of study that owes much to one of history’s greatest minds: Blaise Pascal.

But how exactly do bubbles, tickets, and the work of a 17th-century mathematician come together to shape the New Year’s festivities? Let’s dive deeper into these connections, exploring how the randomness we celebrate today is tied to the enduring legacy of probability.

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Lovely Bubbly!

Champagne: The Joy of Seemingly Random Bubbles

Champagne, the quintessential drink of celebration, owes its festive sparkle to the science of bubbles. When a bottle is uncorked, dissolved carbon dioxide escapes the liquid in the form of bubbles, creating a mesmerizing dance that captivates the eye.

The path of each bubble as it rises is influenced by countless factors: microscopic imperfections in the glass, variations in temperature, and the liquid’s viscosity. This randomness ensures that no two glasses of champagne sparkle exactly alike!

From a scientific perspective, the behavior of these bubbles can be described by probability. Researchers have found that the size and speed of bubbles follow predictable distributions, even though their individual journeys seem random. This interplay of order and chaos mirrors the essence of probability itself—a tool to understand patterns in seemingly unpredictable phenomena.

The randomness of champagne bubbles is more than a scientific curiosity; it embodies the spirit of New Year’s Eve. As we raise our glasses, the unpredictable paths of the bubbles remind us of the uncertainty and spontaneity of life, making each toast a celebration of possibility and chance.

Lovely Bubbly!

From Bubbles to Probability: The Theory Behind the Sparkle

The unpredictable rise of champagne bubbles offers a glimpse into the deeper workings of probability. At first glance, the bubbles seem chaotic and without pattern, much like the events of daily life. Yet, when studied collectively, they reveal predictable trends and distributions—a hallmark of probability theory.

This mathematical framework allows us to move beyond randomness and uncover the rules governing seemingly erratic phenomena. Whether it’s the rate at which bubbles escape a glass or the odds of drawing a winning lottery ticket, probability provides the tools to analyze and understand uncertainty.

It was Blaise Pascal, alongside Pierre de Fermat, who laid the groundwork for modern probability theory in the 17th century. What began as a mathematical inquiry into games of chance evolved into a profound tool for studying the randomness of the world. This connection between the unpredictability of champagne bubbles and the structure of probability sets the stage for the next element of our New Year’s traditions: lotteries.

Blaise Pascal, born June 19, 1623 in Clermont-Ferrand, France.

Blaise Pascal – A Pioneer in Mathematics and Philosophy

Blaise Pascal, a 17th-century French mathematician, physicist, and philosopher, is widely regarded as one of the founding figures of probability theory. His collaboration with Pierre de Fermat in 1654 to solve problems related to gambling games marked the birth of this mathematical field. Their correspondence addressed questions such as how to fairly divide stakes in an interrupted game—a problem now known as the “problem of points.”

The “problem of points” involves determining how to fairly split the stakes of a game if it is halted before its conclusion. For example, consider two players engaged in a game where the first to win three rounds claims the pot. If the game is interrupted with one player having won two rounds and the other one round, how should the pot be divided?

Pascal and Fermat solved this by considering the possible outcomes of the remaining rounds, calculating the proportion of chances each player had of ultimately winning the game. This approach to breaking down complex scenarios into probabilities formed the foundation for modern decision theory, risk analysis and ultimately the online gambling industry.

Pierre de Fermat – A Mathematical Luminary

Pascal’s partner in developing probability theory, Pierre de Fermat, was an equally brilliant mathematician whose work left an indelible mark on mathematics. Fermat is perhaps best known for Fermat’s Last Theorem. Bear with me for a bit o’ mathematics. His theorem (or statement) says:

There are no three positive integers \text{a, b} and \text{c} that satisfy the equation a^n + b^n = c^n for any integer n > 2

That means when n = 1 or n = 2, there are solutions, but not when n is greater than 2.

As n > 2 is infinite, how do you prove Fermat is correct?

(Pythagoras told us about when n = 2 as this relates to his right angled triangles. 3^2 + 4^2 = 5^2 that is 9 = 16 = 25 and 5^2 + 12^2 = 13^2 that is 25 + 144 = 169)

Rendered by QuickLaTeX.com

Although Fermat famously claimed to have a proved his own theorem, the proof was “too large to fit in the margin” of his notebook. The theorem remained unproven for over 350 years until British mathematician Andrew Wiles solved it in 1994.

Fermat’s contributions extended beyond this theorem; he also worked on analytic geometry, calculus, and number theory, laying the groundwork for modern mathematics.

Pascal’s Wager – Betting on God

Does God exist? It’s a good bet!

Pascal’s Wager: Betting on God’s Existence

Pascal was not only a brilliant mathematician, but was also a philosopher. The two subjects blurred into each other when Pascal mused the chances of God’s existence.

Pascal’s Wager addresses the question of whether it is rational to believe in God, even without definitive proof of His existence.

Pascal argued that humans face two choices: to believe in God or not. Each choice carries potential consequences!

  • If God exists and you believe, the reward is infinite (eternal happiness in heaven).
  • If God exists and you do not believe, the loss is infinite (eternal damnation).
  • If God does not exist and you believe, the cost is finite (a life lived in faith without a divine reward).
  • If God does not exist and you do not believe, the gain or loss is also finite.

From this perspective, Pascal reasoned that belief in God is the safer and more rational choice, as it offers the potential for infinite gain with minimal risk.

My Night at Maud’s

The 1969 French film My Night at Maud’s, directed by Éric Rohmer, beautifully captures Pascal’s multifaceted legacy. The movie follows Jean-Louis, a devout Catholic engineer, who engages in a philosophical debate with Vidal, a Marxist lecturer, and Maud, a free-spirited divorcee. The characters discuss topics like faith, morality, and Pascal’s Wager, reflecting on the interplay between reason and belief.

A pivotal scene involves a debate about the rationality of belief in God, directly referencing Pascal’s Wager. The film’s exploration of decision-making under uncertainty echoes Pascal’s work in probability, making it a thought-provoking study of human choices.

This pragmatic approach to faith aligns with the principles of probability, weighing outcomes based on their likelihood and impact. While the Wager does not attempt to prove God’s existence, it provides a framework for making decisions under uncertainty, resonating with themes central to both probability theory and gambling.

Probability is at the heart of the gambling industry

Probability: Fundamental Knowledge for Gambling

Probability and odds are central concepts in the world of gambling, shaping how games are played and how bets are placed.

What is Probability?

Probability is a measure of how likely an event is to occur. It is usually expressed as a number between 0 and 1, where 0 means the event will not happen, and 1 means the event will certainly happen. The formula for calculating probability

(1)   \begin{equation*}\left\text{Probability} = \right\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\end{equation*}

For example, in a fair coin toss, there are two possible outcomes: heads or tails. The probability of getting heads is:

(2)   \begin{equation*}\text{Probability of Heads} = \frac{1}{2} = 0.5\end{equation*}

In gambling, probability helps to assess how likely a specific outcome is in a game. This can apply to rolling dice, drawing cards, or even betting on sports.

Odds are you know this already 🙂

What are Odds?

Odds are a way of expressing the likelihood of an event occurring, but they are presented in a different format than probability. Odds are usually written as a ratio of two numbers. There are two common types of odds:

  1. Fractional Odds: These are often used in sports betting and represent the ratio of the amount you can win relative to the stake. For example, odds of 5/1 mean you can win $5 for every $1 you bet.
    • If you bet $10 at 5/1 odds, you would win $50 if the bet is successful.
  2. Decimal Odds: These are common in European betting and represent the total payout (including your stake) for each dollar bet. For example, decimal odds of 6.00 mean that for every $1 wagered, the total payout will be $6.
    • If you bet $10 at 6.00 odds, you would receive $60 (your $10 stake plus $50 in winnings).
American Odds Format

American odds, also known as moneyline odds, are commonly used in sports betting in the United States. They can appear as either positive (+) or negative (-) numbers, and they indicate how much a bettor can win or needs to bet in order to make a profit. Understanding how American odds work is crucial for anyone involved in sports betting.

Positive American Odds (+)

Positive American odds show how much profit that can be made on a $100 bet. They are typically used to represent underdogs in a match.

Example: +200

  • A bet of $100 on odds of +200 will win $200 if the bet is successful, in addition to getting back the original $100 wagered.
  • In other words, for every $100 bet, the win is $200 (if the bet wins).
  • The formula for calculating the potential payout is:

    (3)   \begin{equation*}\text{Payout} = (\frac{\text{Odds}}{100}) \times \text{Bet}\end{equation*}


    So, with +200 odds:

    (4)   \begin{equation*}\text{Payout} = (\frac{200}{100}) \times 100 = 200\end{equation*}


    The total payout will be $300 (the original bet of $100 plus $200 in winnings).

Negative American Odds (-)

Negative American odds indicate how much you need to bet in order to win $100. They are typically used to represent favorites in a match, as the favored team or player is more likely to win.

Example: -150

  • A bet of $150 on odds of -150 will win $100 if the bet is successful, in addition to getting back the original $150 wagered.
  • In other words, to win $100, the bet would need to be $150.
  • The formula for calculating the potential payout is:

    (5)   \begin{equation*}\text{Payout} = (\frac{100}{\text{Odds}}) \times \text{Bet}\end{equation*}


    So, with -150 odds:

    (6)   \begin{equation*}\text{Payout} = (\frac{100}{\text{150}}) \times 150 = 100\end{equation*}


    The total payout will be $250 (the original bet of $150 plus $100 in winnings).

How to Convert American Odds to Probability

American odds can also be converted into implied probability, which helps you understand the likelihood of an event happening based on the odds.

  • Positive Odds (+): To convert positive odds to implied probability, use the following formula:

    (7)   \begin{equation*}\text{Implied Probability} = \frac{100}{\text{Odds} + 100} \end{equation*}

  • ​Example for +200

    (8)   \begin{equation*}\text{Implied Probability} = \frac{100}{200 + 100} = 0.3333 \text{ or } 33.33%\end{equation*}

  • Negative Odds (-): To convert negative odds to implied probability, use this formula

    (9)   \begin{equation*}\text{Implied Probability} = \frac{\text{Absolute Value of Odds}}{\text{Absolute Value of Odds} + 100} \end{equation*}


    Example for -150

    (10)   \begin{equation*}\text{Implied Probability} = \frac{150}{150 + 100} = 0.6 \text{ or } 60%\end{equation*}

Relationship Between Probability and Odds
Relationship Between Probability and Odds

Probability and odds are closely related, and you can convert between the two. Here’s how:

  1. From Odds to Probability: To convert fractional odds into probability, use the formula:

    (11)   \begin{equation*}\text{Probability} = \frac{\text{Denominator of the odds}}{\text{Denominator of the odds} + \text{Numerator of the Odds}}\end{equation*}


    For example, with odds of 5/1:

    (12)   \begin{equation*}\text{Probability} = \frac{1}{5 + 1} = \frac{1}{6} = 0.1667 \text{ or } 16.67%\end{equation*}

  2. From Probability to Odds: To convert probability into odds, use the formula:

    (13)   \begin{equation*}\text{Odds} = \frac{1 - \text{Probability}}{\text{Probability}}\end{equation*}


    For example, if the probability of an event is 0.25 (25%), the odds would be:

    (14)   \begin{equation*}\text{Odds} = \frac{1 - 0.25}{0.25} = \frac{0.75}{0.25} = \text{3/1}\end{equation*}

House Edge and How It Affects the Game

In most gambling games, the house (casino or bookmaker) has an advantage, known as the house edge. This means that, on average, players are more likely to lose than win. The house edge is built into the odds offered by the casino, making it less likely that a player will win in the long run.

For example, in a game of roulette, the odds of winning a single number bet are 37/1 in European roulette (because there are 37 pockets: 1 to 36 plus a single zero). However, the payout for winning is only 35/1, which gives the house an edge.

House Edge and How It Affects the Game
House Edge and How It Affects the Game

In most gambling games, the house (casino or bookmaker) has an advantage, known as the house edge. This means that, on average, players are more likely to lose than win. The house edge is built into the odds offered by the casino, making it less likely that a player will win in the long run.

For example, in a game of roulette, the odds of winning a single number bet are 37/1 in European roulette (because there are 37 pockets: 1 to 36 plus a single zero). However, the payout for winning is only 35/1, which gives the house an edge.

It’s not all about mathematics

Confucius, the I Ching, and the Wisdom of Randomness

Long before Blaise Pascal developed probability theory, ancient Chinese philosophy explored randomness and its deeper meanings through the I Ching, or the Book of Changes. This revered text, dating back over 3,000 years, provides a framework for understanding life’s uncertainties and the hidden patterns within apparent chaos.

The I Ching and Its Philosophical Roots

The I Ching is based on the concept of constant change, driven by the interplay of opposing forces, yin and yang. These dualities represent the dynamic balance of life—light and dark, order and disorder, certainty and uncertainty. By consulting the I Ching, individuals seek insight into how these forces shape their lives and guide their decisions.

The Yarrow Stalk Method: Randomness with Meaning

Traditionally, the I Ching was used with 50 yarrow stalks. Through a meticulous process of dividing and counting the stalks, users would generate one of 64 hexagrams, each composed of six lines (either broken or unbroken). The randomness in this process mirrors the unpredictability of life events, while the resulting hexagrams offer symbolic guidance on navigating life’s complexities.

Modern interpretations often use coin tosses for simplicity, but the principle remains the same: random outcomes are imbued with meaning through interpretation. This practice highlights a unique perspective on randomness—not as pure chaos, but as a reflection of a deeper order waiting to be uncovered.

Confucian Influence on the I Ching

Confucius, one of China’s most influential philosophers, deeply respected the I Ching and its teachings. He saw it not merely as a tool for divination but as a source of moral and philosophical wisdom. For Confucius, the randomness of the yarrow stalks symbolized the unpredictability of life, while the hexagrams provided a framework for understanding one’s place within the cosmic order.

Through its blend of chance and structure, the I Ching embodies a philosophical approach to randomness that resonates with modern concepts of probability. It reminds us that even in the face of uncertainty, there is potential for insight, growth, and harmony.

Lotteries: Dreams Built on Probability

At the heart of every lottery lies probability. When you buy a ticket, you’re participating in a system governed by precise mathematical rules. For example, the odds of winning the Powerball jackpot are approximately 1 in 292 million. These figures might seem daunting, but they are a direct application of probability theory.

Probability doesn’t just define the chances of winning; it also shapes the design of lotteries. By carefully balancing the size of the jackpot, the cost of tickets, and the odds of winning, organizers create a system that maximizes participation while ensuring profitability. This delicate balance wouldn’t be possible without the mathematical insights pioneered by Pascal and his contemporaries.

Yet, people continue to play, driven by the hope that they might defy the odds. This hope is rooted in the human tendency to dream big—to imagine that, against all statistical likelihood, they could hold the winning ticket. In this way, lotteries capture both the rational and emotional sides of probability, blending mathematical structure with the thrill of possibility.

As we enter the New Year, lottery tickets symbolize more than just a chance at wealth. They represent our willingness to take risks, to embrace uncertainty, and to hope for a brighter future. In the next section, we’ll explore how Pascal’s work not only shaped probability theory but also offered philosophical insights into how we approach life’s uncertainties.

Conclusion: Embracing Chance

As the New Year begins, we find ourselves drawn to traditions that celebrate both the chaos and the order of life. Whether it’s the playful rise of champagne bubbles, the hopeful purchase of a lottery ticket, or the contemplative wisdom of the I Ching, these practices remind us that randomness is not something to fear but something to embrace.

The science of probability offers us tools to understand the unpredictable, yet it also invites us to appreciate the beauty of uncertainty. From Blaise Pascal’s groundbreaking theories to the ancient reflections of Confucius, the study of chance has continually shown us that within randomness lies meaning, and within uncertainty lies opportunity.

As we toast to another year, let us celebrate not only the traditions of the season but also the deeper truths they reveal. For in the unpredictability of life, we find endless reasons to dream, to hope, and to believe in the possibilities of tomorrow!

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