... balls.

Those who understand Italian might form an idea of a real session watching a video of a conference for the general public organized by the University of Roma 3 in June 2016 (http://orientamento.matfis.uniroma3.it/fisincittastorico.php#dagostini) and available on YouTube (https://www.youtube.com/watch?v=YrsP-h2uVU4).

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... cost.

For this purpose this kind of lotteries are preferable to normal bets, although hypothetical and even those with small amount of money (value and amount of money are well known for not being proportional), in order to allow people to freely choose what they consider more credible, without incurring the so called loss aversion bias.

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... to

In this particular case it is clear that `it has to', but in general `it might'. See for example footnote 9 and pay attention that conditional probabilities might be not intuitive and a formal guidance is advised.

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... probable,

Please compare this expression, “the extraction of White becomes more probable”, with “the probability we assign to it”, used above. The former should be, more correctly, “we assign higher probability to the extraction of White”, as it will be clear later. For sake of conciseness and avoiding pedantry, in this paper I will often use imprecise expressions of this kind, as used in every day language.

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... waving.

See e.g. https://www.youtube.com/watch?v=YrsP-h2uVU4 from 48:00 (in Italian).

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... out.

Here is the result with a single line of R code:
> N=5; n=20; i=0:N; pii=i/N; pii^n/sum(pii^n)
[1] 0.000000e+00 1.036587e-14 1.086940e-08 3.614356e-05 1.139740e-02 9.885665e-01
(And, by the way, this is a good example of the importance of a formal guidance in assessing probabilities: according to my experience, after a sequence of 5-6 White, people are misguided by intuition and tend to believe box $B_5$ much more than they rationally should.)

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... get

Here is the R code for the example of 20 extractions resulting in 5 White:
> N=5; n=20; i=0:N; pii=i/N; x=5; pii^x * (1-pii)^(n-x) / sum( pii^x * (1-pii)^(n-x) )
[1] 0.000000e+00 6.968411e-01 2.979907e-01 5.167614e-03 6.645594e-07 0.000000e+00
(Note how using this code we can focus on the essence of what it is going on, instead of being `distracted' by the math of the normalization.)

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...Laplace:

In the light of Brecht's quote by Galileo you might be surprised to find quite some quotes in this paper. But there are books and books.

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... likely,

 This would have been the correct answer to a different question: probability of White from a box taken at random among boxes $B_{1-5}$, that is $B_?^{(1-5)}$. Ruling out $B_0$ by hand at the very beginning is quite different from ruling it out as a consequence of the described experiment. The status of information is different in the two cases and also the resulting probabilities will usually be different! [Please note that a different state of information might change probability, but not necessarily it does. For example $P(\mbox{W}^{(1)}\,\vert\,I) = P(\mbox{W}^{(11)}\,\vert\,5\mbox{B},5\mbox{W},I)$ just by symmetry. Conditioning is subtle!]

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... get

 Here is the numerical result obtained with R:
> N=5; i=0:N; pii=i/N; ( PBi = pii/sum(pii) ); sum( pii * PBi )
[1] 0.00000000 0.06666667 0.13333333 0.20000000 0.26666667 0.33333333
[1] 0.7333333

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... it.

Curiously, for strict frequentists the probability that $B_?$ contains $i$ white balls makes no sense because, they say, either it does or it doesn't.

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....

 The notation used above is consistent with this statement, in the sense that the conditions appearing in $P(B_i\,\vert\,I)$, $P(B_i\,\vert\,\mbox{W}^{(1)},I)$ and $P(B_i\,\vert\,\mbox{W}^{(1)},\mbox{W}^{(2)},I)$ can be seen seen as $I_S(t)$ evolving with time.

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... `objective'.

It is curious to remark that there are, or at least there were, also Bayesians `afraid' of subjective probability (7).

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...,

 Note also this very last statement, to which we shall return at the end of the paper.

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....

 As a real example, in my talk at MaxEnt 2016 I analyzed the football match France-Portugal, played right on the first day of the workshop, so that everybody (interested in football) had fresh in their minds the reaction of fans of the two teams, as shown on TV, and also that of people in pubs in Ghent (slides are available at http://www.roma1.infn.it/~dagos/prob+stat.html#MaxEnt16_2).

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... although

What Hume says about probability reminds me of the famous reflection by Augustine of Hippo about time: “Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio.“ - “What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.” (https://en.wikiquote.org/wiki/Augustine_of_Hippo.) Indeed, as a creature living in a hypothetical Flatland has no intuition of how a 3D world would be, so a hypothetical intelligent humanoid `determinoid,' living in a (very boring) world in which all phenomena happen with extreme regularity, would have not developed the concept of probability.

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... black.

 The exact number of $P(\mbox{W}^{(5)}\,\vert\,4\mbox{W},I)$ is 90.4%, as it can be easily checked with R:
> N=5; n=4; i=0:N; pii=i/N; ( PBi=pii^n/sum(pii^n) ); sum(pii * PBi)
[1] 0.00000000 0.00102145 0.01634321 0.08273749 0.26149132 0.63840654
[1] 0.9039837

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... event.”

The second position, popularized by Einstein's “God does not play dice”, is related to the so-called Laplace Demon, “An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” (2)

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... neutral).

The branching ratios of K$^+$ into the two `channels' are $\mbox{BR}(\mbox{K}^+ \rightarrow \mu^+\nu_\mu) = (63.56\pm 0.11)\%$ and $\mbox{BR}(\mbox{K}^+ \rightarrow \pi^+\pi^0) = (20.67\pm 0.08)\%$ (12).
By the way, I do not think that Quantum Mechanics needs special rules of probability. There the mysteries are related to the weird properties of the wave function $\psi(x,t)$. Once you apply the rules - “shut up and calculate!” has been for long time the pragmatic imperative - and get `probabilities' (in this case `propensities', as we shall see) all the rest is the same as when you calculate `physical probabilities' in other systems. Take for example the brain-teasing single photon double slit experiment (see e.g. https://www.youtube.com/watch?v=GzbKb59my3U). From a purely probabilistic point of view the situation is quite simple. Applying the rules of Quantum Mechanics, if we open only slit $A$ we get the pdf $f_A(x\,\vert\,A,I)$; if we open only $B$ we get $f_B(x\,\vert\,B,I)$; if we open both slits we get $f_{A\& B}(x\,\vert\,{A\& B},I)$. Why should $f_{A\& B}(x\,\vert\,{A\& B},I)$ be just a superposition of $f_A(x\,\vert\,A,I)$ and $f_B(x\,\vert\,B,I)$? In fact within probability theory there is no rule which relates them. We need a model to evaluate each of them and the best we have are the rules of Quantum Mechanics. Once we have got the above pdf's all the rest follows as with other common pdf's. In particular, if we get e.g. that $f_A(x_1\,\vert\,A,I) >> f_{A\& B}(x_1\,\vert\,{A\& B},I)$ we believe that a photon will be detected `around' $x_1$, if we open only slit $A$, much more than if we open both slits. And, similarly, if we plan to repeat the experiment a large number of times, we expect to detect `many more' photons `around' $x_1$ if only slit $A$ is open than if both are. That's all. A different story is to get an intuition of the rules of Quantum Mechanics.

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... particle.

I like, as historian Peter Galison puts it: “Experiments begin and end in a matrix of beliefs. ...Beliefs in instrument type, in programs of experiment enquiry, in the trained, individual judgments about every local behavior of pieces of apparatus.” (13) Then beliefs are propagated within the scientific community and then outside. But, as recognized, methods from `standard statistics' (first at all the infamous p-values) tend to confuse even experts and spread unfounded beliefs through the scientific community as well as among the general public (4,5), that in the meanwhile is developing `antibodies' and is beginning to mistrust striking scientific results and, I am afraid, sooner or later also scientists and Science in general.

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...preference)

I have no strong preference on the name, and my propensity in favor of `propensity' is because it is less used in ordinary language (and despite the fact that this noun is usually associated to Karl Popper, an author I consider quite over-evaluated).

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...physical

Note the extended meaning of `physical', not strictly related to Physics, but to `matters of fact' of all kinds, including for example biological, sociological or economic systems believed to have propensities to behave in different ways.

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....

I had heard that this apparent obvious statement goes under the name of Lewis' Principal Principle (see e.g. http://plato.stanford.edu/entries/probability-interpret/). Only at the late stage of writing this paper I bothered to investigate a little more about that `curious principle' and found out Lewis' Subjectivist's Guide to Objective Chance (14), in which his very basic concepts, outlined in a couple of dozen of lines at the beginning of the article, are amazingly in tune with several of the positions I maintain here.

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...:

It becomes now clear the meaning of Equation (7), which we can rewrite as

$$\begin{eqnarray*} f(p\,\vert\,n,x,I) & \propto & p^x\,\left(1-p\right)^{n-x}, \end{eqnarray*}$$

having assumed a continuity of propensity values, and having started our inference from a uniform prior, that is $f(p\,\vert\,I) = 1$.
The normalized version of the above equation is

$$\begin{eqnarray*} f(p\,\vert\,n,x,I) & = & \frac{(n+1)!}{x!\,(n-x)!}\,p^x\,\left(1-p\right)^{n-x}. \end{eqnarray*}$$

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... presently),

Here is, for example, what David Lewis (see Footnote 23) writes in Ref. (14) (italics original): “Carnap did well to distinguish two concepts of probability, insisting that both were legitimate and useful and that neither was at fault because it was not the other. I do not think Carnap chose quite the right two concepts, however. In place of his `degree of confirmation', I would put credence or degree of belief; in place of his `relative frequency in the long run', I would put chance or propension, understood as making sense in the single case.” More or less what I concluded when I tried to read Carnap about twenty years ago: his first choice means nothing (or at least it has little to do with probability); the second does not hold, as I am arguing here.

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... `trials'),

To make it clear, what is important to is that $p$ is (about) the same, and that our assessments are independent. It does not matter if, instead, the events have a different meaning, like e.g. tails tossing a coin, odd number rolling a die, and so on. The emphasized `about' is because $p$ itself could be uncertain, as we shall see later. In this case we need to evaluate the expectation of $f_n$ taking into account the uncertainty about $p$.

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....

Related to this there is the usual confusion between a probability distribution and a distribution of frequencies. Take for example a quantity that can come in many possibilities, like in a binomial distribution with $n=10$ and $p=1/2$. We can think of repeating the trials a large number of times and then, applying Bernoulli's theorem to each of the eleven possibilities, we consider it very unlikely to observe values of relative frequencies in each `bin' different from the probabilities evaluated from the binomial distribution. This is why we highly expect - and we shall be highly surprised at the contrary! - a frequency distribution (`histograms') very similar in shape to the probability distribution, as you can easily `check' playing with
n=10000; x=rbinom(n, 10, 0.5); barplot(table(x)/n, col='cyan')
barplot(dbinom(0:10,10,0.5), col=rgb(1,0,0,alpha=0.3), add=TRUE)
That's all! Nothing to do with the “frequency interpretation of probability”, or with the “empirical law of Chance” (see Footnote 28).

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....”

Obviously, if you make an experiment of this kind, tossing regular coins or dice a large number of times, you will easily find relative frequencies of a given face around 1/2 or 1/6, respectively as simulated with this line of R:
p=1/2; n=10^5; sum( rbinom(n, 1, p) ) / n
But it is just because, in the Gaussian large number approximation, $P(\vert f_n-1/2\vert > 1/\sqrt{n}) = 4.6\%$, and therefore $f_n$ will usually occur around $1/2$ [although all $(n+1)$ values between 0 and 1 are possible, with probabilities $P(f_n=x/n) = 2^{-n}n! \left(x!(n-x)!\right)^{-1}$]. Not because there is a kind of `law of nature' - “legge empirica del caso”, in Italian books, i.e. “empirical law of Chance” - `commanding' that frequency has to tend to probability, thus supporting the popular lore of late numbers at lotto hurrying up in order to obey it. In the scientific literature and in text books, not to speak about popularization books and article, it should be strictly forbidden to call `laws' the results of asymptotic theorems, because they can be easily misunderstood. [For example we read (visited 11/11/2016) in https://it.wikipedia.org/wiki/Legge_dei_grandi_numeri that “the law of large numbers, also called empirical law of chance or Bernoulli's theorem [...] describes ...” (total confusion! - see also https://en.wikipedia.org/wiki/Law_of_large_numbers and https://en.wikipedia.org/wiki/Empirical_statistical_laws).]

Moreover, it should be avoided to teach that e.g. probability 1/3 means that something will occur to 1/3 of the elements of a `reference class', i) first because a false sense of regularity can be easily induced in simple minds, which will then complain that the “the probabilities were wrong” if no event of that kind occurred in 9 times; ii) second because such `reference classes' might not exist, and people should be trained in understanding degrees of belief referred to individual events.

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...analogous

$E^{(1)}$ is the success in the first trial, $E^{(2)}$ the success in the second trial, and so on. Speaking about “the realization of the same event” is quite incorrect, because events $E^{(i)}$ are different. They can be at most analogous. We indicate here, instead, by $E$ the generic future event of the kind of $E^{(1)}$-$E^{(n)}$, i.e. for example $E=E^{(n+1)}$.

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... past.

 It is a matter of fact that, because of evolution or whatever mechanism you might think about, the human mind always looks for regularities. This is how Hume puts it (italics original): “Where different effects have been found to follow from causes, which are to appearance exactly similar, all these various effects must occur to the mind in transferring the past to the future, and enter into our consideration, when we determine the probability of the event. Though we give the preference to that which has been found most usual, and believe that this effect will exist, we must not overlook the other effects, but must assign to each of them a particular weight and authority, in proportion as we have found it to be more or less frequent.” (9)

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... `better'

To get an idea, repeat several times the following lines of R code which simulate n extractions with re-introduction from box ri, calculate the number of White, infer the probability of the box compositions, and finally evaluate the probability of a next White and compare it with the relative frequency. There is no miracle in the result, it is just that the probabilistic formulae are using all available information in the best possible way:
N=5; i=0:N; pii=i/N; ri=1; n=100; s=rbinom(n,1,pii[ri+1]); ( x=sum(s) )
( PBi = pii^x * (1-pii)^(n-x) / sum( pii^x * (1-pii)^(n-x) ) )
cat(sprintf("P(W|sequence) = %.10f; x/n = %.4f \n", sum( pii * PBi ), x/n))

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... time.

I would like to make a related comment on another myth concerning the scientific method, according to which “replication is the cornerstone of Science”. This implies that, if we take this principle literally, much of what we nowadays consider Science is in reality non-scientific (can we repeat measurements concerning a particular supernova, or two particular black holes merging with emission of gravitational waves?). And if you ask, they will tell you that this principle goes back to none other than Galileo, who instead wrote(15) that “The knowledge of a single effect acquired by its causes opens our mind to understand and ensure us of other effects without the need of doing experiments” (“La cognizione d'un solo effetto acquistata per le sue cause ci apre l'intelletto a 'ntendere ed assicurarci d'altri effetti senza bisogno di ricorrere alle esperienze”). Doing Science is not just collecting (large amounts of) data, but properly framing them in a causal model of Knowledge.

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... experiment.

What is nice in this practical session, instead of abstract speculations, is that the people participating in the discussion have developed their degrees of beliefs, and therefore, when the box is taken away, they cannot say that what they were thinking (and feeling!) is not valid anymore.

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... uncertainty.

See e.g. Feynman's quote at the end of the paper.

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... innocent?

If you worry about these issues, then you might be interested in the Innocence Project, http://www.innocenceproject.org/.

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... box.

Note that many statements concerning scientific and historical `facts' are of this kind.

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... circle.”

See e.g. https://developer.android.com/reference/android/location/Location.html#getAccuracy()

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... bet.

Here is how Laplace reported his uncertainty on value of the mass of Saturn got by Alexis Bouvart: “His [Bouvard] calculations give him the mass of Saturn as 3,512th part of that of the sun. Applying my probabilistic formulae to these observations, I find that the odds are 11,000 to 1 that the error in this result is not a hundredth of its value.” (2) That is $P(3477 \le M_{Sun}/M_{Sat} \le 3547\,\vert\,I(\mbox{Laplace})) = 99.99\%\,.$ Note how the expression “the odds are,” indicates he was talking of a fair bet, viz. a coherent bet. Moreover it is self evident that such a bet cannot be, strictly speaking, settled, but it rather had an hypothetical, normative meaning. (And Laplace was also well aware of the non linearity between quantity of money and its `moral' value, so that a bet with such high odds could never be agreed in practice and it was just a strong way to state a probability.)

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... fact,

“If we were not ignorant there would be no probability, there could only be certainty. But our ignorance cannot be absolute, for then there would be no longer any probability at all. Thus the problems of probability may be classed according to the greater or less depth of our ignorance.” (18)

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... probabilities,

Italians might be pleased to remember Dante's “Cred'io ch'ei credette ch'io credesse che ...” (Inf. XIII, 25), expressing beliefs of beliefs of beliefs (“I believe he believed that I believed that...”), roughly rendered in verses as “He, as it seem'd, believ'd, that I had thought [that]...” (https://www.gutenberg.org/files/8789/8789-h/8789-h.htm#link13).

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... questions

For example we can ask the range of virtual coherent bets one could accept in either direction, or `calibrate' probabilistic judgements against boxes with balls of different colors (or other mechanical or graphical tools).

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