Article – The Father of Probability

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

The Father of Probability

Blaise Pascal

His Legacy in Online Gambling

by Ian Sherrington

December 2024

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

Probability is the life-blood of our industry

Gambling by definition involves probability – the chances of something happening, or not.

As a fundamental of the gambling industry, an understanding of probability is essential!

Here follows is a brief summary of probability and it’s definer, Blaise Pascal,

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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 $25 + 144 = 169$ )

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.

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: A Game-Changer for Gambling

Probability theory, as pioneered by Pascal, provides a framework for understanding chance and uncertainty. At its core, probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Key concepts include:

Independent Events: These are events where the outcome of one does not affect the other, such as flipping a coin or rolling a die. Understanding independence is crucial in games like blackjack, where the probability of a specific card being drawn depends on prior draws.
Expected Value (EV): This represents the average outcome of a random event over many trials. For example, in roulette, the EV can help players understand the long-term fairness of a bet by comparing potential payouts to the probabilities of winning.
Variance: This measures the spread of possible outcomes. A game with high variance, like slots, may have large payouts but less frequent wins, whereas a low-variance game, like baccarat, offers smaller, more consistent payouts.

By applying these principles, players can assess risks and rewards with greater precision, and operators can design games with controlled odds and payouts. For instance, the odds of hitting a royal flush in poker are roughly 1 in 649,740, demonstrating how probability governs the frequency of rare events in gambling scenarios. At its core, probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Key concepts in probability include:

Independent Events: Events where the outcome of one does not affect the other, such as flipping a coin.
Expected Value: A measure of the average outcome over many trials, used to determine the fairness or profitability of a bet.
Variance: The degree of variation in outcomes, which helps gauge risk.

These principles revolutionized gambling by enabling players and operators to assess risks and rewards with mathematical precision. For example, in roulette, the probability of landing on a specific number is 1/37 in European roulette or 1/38 in American roulette. Similarly, poker players calculate probabilities to determine the likelihood of drawing winning hands.

From Pascal’s Ideas to Online Gambling

The advent of online gambling has amplified the relevance of probability theory. Algorithms governing digital platforms rely heavily on probabilistic models to simulate randomness, calculate odds, and ensure fairness. For instance, Random Number Generators (RNGs) are central to slot games and online roulette, ensuring unbiased results by generating sequences of numbers that mimic true randomness.

Similarly, Monte Carlo simulations are used in some betting platforms to model the likelihood of various outcomes by running multiple simulations of possible scenarios. These tools, combined with machine learning algorithms, help operators fine-tune odds and predict player behavior, ensuring a balance between fairness and profitability. For example:

Random Number Generators (RNGs): RNGs use complex algorithms to create outcomes that mimic true randomness, ensuring that games like online slots or blackjack are unbiased. For instance, an RNG might ensure that the probability of hitting a jackpot in a slot game aligns with advertised odds.
Betting Odds: Online sportsbooks use probability to set odds for various events, such as the outcome of a soccer match. A team with a 25% chance of winning might have odds of 3:1, influencing how much a bettor stands to win.
Risk Management: Operators leverage probabilistic models to manage risks, predict trends, and ensure profitability. For example, casinos analyze the expected value of each game to maintain an edge over players.

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Understanding Probability and Odds in Gambling

Probability and odds are central concepts in the world of gambling, shaping how games are played and how bets are placed. Understanding these concepts can give players a better grasp of the chances of winning and losing, and help them make more informed decisions.

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.

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:

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.
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

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

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

Conclusion

Understanding probability and odds is essential for anyone interested in gambling Knowing the likelihood of different outcomes helps make better decisions.

Blaise Pascal’s contributions to probability theory revolutionized our understanding of chance, laying the foundation for modern gambling practices. His philosophical musings continue to inspire discussions about risk, faith, and decision-making. In today’s online gambling industry, his legacy endures, blending mathematical rigor with profound reflections on human behavior and belief.

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