## - What is probability theory and Bayesian inference how does it work?

## - How does Bayes Theorem work in practice?

## - Conditional probability in soccer betting

## - Bayesian inference in gambling

In the early 2000s, students delving into Probability Theory were introduced to the notion of "Inverse probability". Fast forward to today, this concept is more commonly referred to as Bayesian probability among scholars.

Bayesian probability is articulated as a perspective where probability does not merely signify the occurrence frequency of an event but embodies a level of personal conviction or understanding regarding a matter. Bayesian inference is a statistical inference approach that employs Bayes' theorem to refine the probability of a hypothesis with the advent of additional evidence or information. Simply put, Inverse probability offers a more rational methodology within the scope of probability theory.

The term Inverse probability is now considered archaic, previously used to denote the probability distribution of a variable that is not observed. Thus, Bayesian inference in gambling merges logical-mathematical analysis, subjective probability, and qualitative assessment aiming to gain a competitive advantage. This inference is rooted in the Bayes Theorem, established by Thomas Bayes, which addresses conditional probability, enabling us to tackle the sports betting markets or any investment venture with it.

## Probability theory in the sports betting markets

Instinctively, we're all trying to harness probability theory within the sports betting domain to identify a competitive edge. The more astute bettors consciously apply these principles, while the pinnacle oddsmakers incorporate complex conditional probability concepts for the most precise market pricing, considering numerous variables.

Consider the following example from my publication "Hypnotised by Numbers," illustrating the application of conditional probability and Bayes Theorem in deriving precise soccer betting probabilities. Before diving into that, however…

## What is the Bayes Theorem?

P(A | B) = P(B | A) *P(A) / P(B)

The Bayes Theorem is a formula that calculates conditional probability, stating: the probability of A happening, given B, is equal to the probability of B happening, given A, times the probability of A, all divided by the probability of B.

## How can we apply conditional probability in soccer betting?

In the 2021-22 Premier League season, Liverpool showcased dominance, narrowly missing the championship to Manchester City, with star player Mohamed Salah in exceptional form.

Some pivotal stats include:

Liverpool secured victories in 74% of their 2021 EPL matches (A).

Salah scored in approximately 55% of matches when adjusted for minutes played and starts (B).

Across an adequate sample size for this period, Salah scored in 60% of Liverpool's winning matches (B|A).

What, then, was the likelihood of Liverpool winning assuming Salah scores?

Utilizing the Bayes Theorem formula or a Bayes calculator like the one available on Sharp Sports Bettors, we find the probability of Liverpool winning with Salah scoring adjusts to an average of 80.72%.

**P (A|B) = 60*74 / 55** -> **P (A|B) = 4,440 / 55** -> **P(A|B) = 80.72%**

However, in the betting markets, odds compilation depends on the specific market and may not always accurately reflect the nuances, such as Salah's impact, considering each selection individually. The odds largely rely on "big data" or what's known as Frequentist probabilities. Employing Bayesian reasoning with small data samples against these big data averages can give us an upper hand in sports betting markets, favoring the mean over the median and leveraging conditional probability.

**“Big data contains greater variety in larger volumes and with more velocity. Put simply, big data is larger, more complex data sets, especially from new data sources.”**

Marco Blume, ex-Trading Director at 7x7Bets, emphasized the importance of qualitative analysis and Bayesian-style thinking in data science and pricing at 7x7Bets. Blume, in a podcast, discussed dealing with "known knowns, known unknowns, and unknown unknowns" in reference to uncertainty in betting markets, especially with small samples and events lacking solid data. He explained how Bayesian inference could be utilized to estimate accurate probabilities.

Betting markets are primarily based on frequentist probabilities and mathematical heuristics. Unlike the Frequentist approach, which does not estimate probabilities of unknown events, Bayesian methodology in market pricing incorporates empirical observations and hypothesis testing to establish an efficient prior price. Frequentist methods rely on data analysis implied probabilities in odds compilation, while true probabilities in sports betting remain uncertain, as opposed to fixed variables in games like poker, roulette, or blackjack with multiple decks.

Enrico Fermi, a physicist, applied qualitative analysis and subjective judgment to estimate solutions to questions whose answers might not be intuitively obvious or documented. This analytical approach could be employed in pricing sports betting markets, like the Conor McGregor vs. Floyd Mayweather fight, where prior data was absent.

## The Monty Hall Puzzle

You've probably encountered the tale of Marilyn vos Savant—deemed the most brilliant woman at the time—when she correctly solved a conditional probability puzzle in her column, only to face backlash from thousands, including PhD-holding mathematicians. This puzzle later earned fame as the Monty Hall problem.

The Monty Hall problem illustrates the complexity of conditional probability, showcasing how a single piece of crucial information can significantly alter probabilities. The key "conditional" here is the game show host's knowledge of what's behind each door, a fact that challenges many of the world's brightest minds.

When it comes to pricing markets based on such principles, soft bookmakers often overlook conditional or subjective variables. This oversight presents a strategic opportunity to outmaneuver betting markets and the inherent margins, a topic we'll delve into in a subsequent piece. Despite the perceived efficiency of crowd wisdom and betting exchange markets at close—provided they're liquid—flaws exist due to the foundational concepts these markets neglect.

**"It is better to be roughly right than precisely wrong" - John Maynard Keynes**

## Rock-Paper-Scissors Theory

Consider the Rock-Paper-Scissors game, where both you and your opponent randomly choose each option one-third of the time. Theoretically, this should result in a break-even outcome over time. However, if you detect a pattern in your opponent's "random" choices, such as choosing Rock every fifth turn, you can exploit this pattern for a significant advantage.

If you adjust your strategy based on this observation, choosing Paper every fifth round to counter their Rock, your expected positive return jumps to around 60/40, equating to a 20% investment return, despite both parties maintaining a one-third selection rate overall.

## Applying Probability Concepts at the Poker Table

The same principle applies in poker. Suppose you bluff 30% of the time on certain river boards, adhering to a game theory optimal (GTO) strategy. An opponent who notices a tell—say, your nose twitches during those bluff attempts—gains a substantial expected value (EV) by accurately calling or folding against your river bets in each scenario.

## Which Poker Hand is Superior?

Presenting a Bayesian and a Frequentist with the question of which poker hand is superior—A8 off-suit versus A4 suited—yields different perspectives. The Frequentist approach, based on extensive simulations, shows A8 winning 51% of the time, A4 suited 29%, with a 20% chance of a tie. However, the Bayesian viewpoint favors A4 suited due to its playability and flexibility, allowing for easier escapes from dominated situations and opportunities for flush and straight draws, enabling creative play and semi-bluffs.

Skilled players find A4 suited-type hands more profitable than A8 off-suit, often avoiding mid-aces in favor of aggressively playing low wheel suited aces, leading to winning large pots or minimizing losses, contrasting with A8's smaller wins or potential for trouble, despite its technical superiority.

## Probability Theory in Betting and Gambling: A Summary

"There are few who did not learn the probabilistic theory and probabilistic thinking, born to be the one who always acts on probability outcomes" - Catherine Wang

Probability quantifies the chance of an event's occurrence. It's foundational to sports betting and gambling, enabling the estimation of different outcomes' likelihoods through logical and mathematical reasoning.

Conditional probability, the likelihood of an event occurring given another influencing event has already happened, is crucial for optimizing strategies across sports betting, gambling, and investment, reflecting the essence of probabilistic thinking in decision-making.

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