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Introduction – About the Author
Index of Cited Names
Table of Contents
Thank you for buying this ebook by Paolo MancaSubjective Probability: the Only Kind Possible
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3. The simple lottery
4. Composite lotteries
5. Kolmogoroff’s formal model
6. Probability: the only possible definition
7. Conditional probability and Bayes’ Theorem
8. Random variables, mean value, risk
9. Independent random variables,exchangeable random variables
10. Jakob Bernoulli’s Theorem
11. The Frequentist “theory” of probability
12. Probability and statistics
13. Some so-called “Paradoxes of Probability”
14. Gambling and the theorem of the ruin of the gambler
15. Attempts to “objectify” probability.
16. The limits of probability and Fuzzy Logic
Plea in mitigation
In the context of a non-deterministic view of the world, probability is a number which reflects the degree of confidence someone has that a given statement or event is true, based on the information they have. Probability and probability calculus, on the other hand, are essential tools for those who wish to recognise uncertainty and manage it responsibly.
This presentation of subjective probability (the only one possible) will enable even those who are not experts to acquire the correct knowledge of an instrument which is essential and necessary for dealing with the world around us.
The text takes only one hour to read, but it will help you to avoid many mistakes and enable you to understand the origin of those you might have made in the past.
The reader will also be able to appreciate many funny examples and paradoxes such as this one: statistics tell us that 20% of motorway car accidents are caused by drivers with high blood alcohol levels. It can then be derived that 80% of accidents are caused by sober drivers. Therefore, we should supply alcohol to those who drive on motorways!
* * *
Paolo Manca professor of Probability Calculus and Chair of Financial Mathematics at the University of Pisa, Director of the Masters Program of Finance and Financial Markets, currently works as a financial consultant, specialized in derivative valuations and savings protection. He is president of the association cerfidi (Centro Studi di Finanza e Diritto) and author of numerous publications in financial economics.
This book was prompted by the contents of some of the textbooks on probability and statistics that are on the reading lists even today, in 2016 AD1, of university courses, from reading entertaining drivel floating about on the internet, and by the survival of internet sites and television adverts that sell “foolproof” methods of winning at gambling and making a fortune on stock market.
I would never have believed it possible that such nonsense about probability could still be said and read, but there we are.
Although the fundamentals of the intrinsic nature of probability were laid out in the 1930s, especially by the Italian genius Bruno de Finetti (as always, appreciated more abroad than at home), and excellently restated and enlarged upon by distinguished scholars, it appears that their work was all in vain.
Being surprised that his ideas, simple yet brilliant, are still ignored, I felt obliged to set out, with all due humility, Bruno de Finetti’s thinking on probability, trying, as far as possible, to avoid technicalities, in order to be understood not only be those familiar with the field but also by those who are not familiar with mathematics and its jargon.
It is, perhaps, a foolhardy mission, since I will doubtless be criticised by experts and non-experts alike. The former will object to certain simplifications that, I myself acknowledge, to be rather insouciant; the latter will object to the more complex reasoning that I have been forced to resort to from time to time.
I think the great debates of the past centuries: the Ptolemaic system and the Copernican, the theories of evolution of species and creationist, to theories of relativity of Galileo and Einstein, nonetheless I am surprised to note that, even today, there is still debate about the nature of probability.
Yet probability can be expressed in this single sentence:
In the context of a non-deterministic view of the world, (which seems more appropriate than a rigid determinism) probability is a number that explains the degree of confidence that a given subject expresses about the truth of a given statement/event, whilst the probability calculus (PrC) is a set of rules and theorems that allow us to draw logical inferences from initial evaluations of probability.
1 The citations would be endless.
The term uncertainty is the opposite of the term certainty; we will use the word uncertainty interchangeably with the words casualness, chance, aleatory (randomness) and stochasticity, underlining the conceptual difference by using terms such as inaccuracy, vagueness and indeterminacy.
Chance (in Italian caso) invokes the Latin casus (fall) used in the sense “that which falls upon him”; aleatory (random) invokes the term alea which in Latin means die (from which, the famous words of Caesar: alea iacta est –the die is cast–); stochastic comes from the Greek stocasticos which literally means conjectural.
Unfortunately, in our society, much of the organisation of knowledge carries deterministic connotations and uncertainty is seen as disorder, irregularities as uncomfortable. Yet uncertainty should be an essential physiological feature in many areas and in many models of both classical sciences and social sciences.
Uncertainty is something that characterises the whole of our lives and the world around us, and the need to make decisions in unpredictable conditions occurs daily.
Sometimes the lack of certainty is inherent because it relates to future events. Sometimes it relates to the lack of available information, which could be obtained, but which is not, for reasons of the cost or the length of time obtaining the information would involve.
When we need to make a decision quickly, on issues of limited significance, we trust in common sense and often in instinct, and this is inevitable.
However, there are important issues and decisions that can have a profound impact on the course of our lives and for which we must proceed in the most well informed way possible: it is here that probability and probability calculus (PrC) are an essential tool for those who wish to recognise uncertainty and manage it responsibly.
We must therefore, as mentioned above, reject the deterministic view of the world around us and instead embrace probability and the probability calculus offered by science as appropriate tools for reasonable people to use to understand and manage rationally a large part of the considerable uncertainty that surrounds us.
Reflect upon the criteria that you used to decide how and where to invest your savings, how you insure against adverse events (death, theft, fire, etc.), how we choose a partner, where to go on holiday...
Try to answer the following questions:
Why in poker is four aces worth more a full house, and, in general, why is four of a kind worth more than full house?
Why, in the lottery, is having five numbers come up worth more than having two numbers come up?
If I choose two lottery numbers and they come up, how much should I win?
In roulette, the house has a small advantage (the zero) but also many expenses (premises, croupiers, management...) yet still makes a significant profit. How come?
What comment would you make on the following assertions?
You have to risk more to earn more.
You shouldn’t put all your eggs in the same basket.
Weather forecasts are very unreliable.