In a well-known probabilistic riddle is asked to imagine that a certain Mr. Rossi by saying, "I have two children, one of them is male, and to calculate the probability that the ' Another son of Mr. Rossi is male.
This riddle book appeared in Martin Gardner sullo Scientific American dedicata ai giochi matematici, in un articolo sulle difficoltà concettuali della probabilità. Gardner infatti si meravigliava del fatto che molte persone intelligenti non solo davano la risposta sbagliata all'enigma (50%), ma non riuscivano ad accettare il fatto che fosse sbagliata nemmeno di fronte alla risposta corretta e alla sua spiegazione.
Infatti, secondo la sua spiegazione, le possibilità con due figli sono quattro in tutto: MM, MF, FM, FF. Escludendo la quarta possibilità in virtù dell'informazione che il signor Rossi ci ha dato, fra le tre rimanenti ne rimane una sola con due maschi, quindi la probabilità è 1/3. Ovvero, una volta presa la popolazione of all those who have two children, one male, only one third of these will have both boys.
Ironically, however, Martin Gardner would have to amend his statement in the next number in the phone book, after many letters of protest received. This is because the people who did not accept the answer presented as correct, it turned out after careful meditation, then they had a point.
inductive inference (probability) can be of two types: direct or reverse. The direct inference is one that tries to infer the characteristics of the sample to those of the population (we have an urn with 50 black balls and 50 white balls, what is the probability che estraendo una pallina essa risulti bianca?). L'inferenza inversa è quella che cerca di inferire le caratteristiche della popolazione da quella del campione: abbiamo un'urna con 100 palline di colore ignoto, se ne estraiamo dieci bianche, qual è la probabilità che tutte le palline nell'urna siano bianche?
Nell'indovinello del signor Rossi è presente un'ambiguità, per cui in realtà non sappiamo esattamente cosa ci viene chiesto. Non è affatto scontato che ci troviamo di fronte a un caso di inferenza diretta, nel quale ci viene chiesto di calcolare la probabilità richiesta semplicemente considerando le caratteristiche (già note) di una data popolazione. La difficoltà, invece, è proprio to understand what population should be considered part Mr. Rossi. At that all people with two children in a male, as suggested by Gardner's solution? And why should not we instead consider the population belonging to all people with two children, of any sex?
We do not know in what capacity Mr. Rossi has provided us with that information, if that is wanted us to calculate the relative frequency of a particular characteristic in a given population (that of fathers with two children in a male), or if he was asking us to provide an estimate of our confidence that the other child is male (ie, how much to bet on that possibility). And the two things, though on this there is some conspiracy of silence, are very different.
no inference is that the reverse is not mathematically calculable. In the example of the urn and balls (100 balls of unknown color, or extract ten white), there is a precise formula, which is given by Bayes' theorem . The problem is that this formula must be used to force use of arbitrary assumptions about the so-called "prior probability", not taken from any observation, but simply postulated (perhaps by using the Laplacian "principle of indifference").
must assume, for example, that all the different distributions of color in the urn are equally likely a priori (100 white and 0 black, 99 white and 1 black, 98 white and 2 black, etc.), then calculate, using Bayes' theorem, as these probabilities vary as a function of the extractions done. But some people might dispute this principle and believe that some distributions are more likely. For example, it is clear that in a series of coin tosses are the most likely combinations that provide a balance between heads and tails (50 heads and 50 crosses) to those involving cross or only one witness, and you do not see why a similar principle can not apply to the case of the urns and balls.
The riddle of Mr. Rossi serves to bring into the open a clash between two different philosophical views, that "objectivity" (or "frequentist") and "subjective" about the chances. For some, the probability is something that concerns only the direct inference, and can be applied only when we have some objective data (statistics on mortality among smokers, for example, can help us calculate the probability of getting cancer). The reverse inference is not legitimate to use instead of calculating the odds.
For others, however, the probability is something inherently subjective, it is not simply the degree of confidence that a certain person has nell'occorrere of a certain event. It is true, it can be modified by experience (I would be irrational if the continued take of certain events do not change my choice, even take their future), but the subjective element can never be completely eliminated from the data. The best-known spokesman for the subjectivist conception of probability, by the way, was a great Italian mathematician, Bruno De Finetti, one of the genes that have trod the soil of our homeland.
The fact, however, is that cases of induction in scientific reasoning, or at least as regards the formulation of theories and the discovery of new scientific laws, always about the negative inference. The direct inference is required in practice a calcolare le probabilità di uscita di una combinazione di numeri al superenalotto, o di azzeccare un numero alla roulette. Solo casi, cioè, di "probabilità addomesticata", nei quali la popolazione di riferimento è nota perché da noi decisa e posta sotto il nostro stretto controllo.
Per quella che Nassim Nicholas Taleb chiama " fallacia ludica " molti testi divulgativi di teoria della probabilità tendono a concentrarsi solo sui casi addomesticati, dando una visione parziale e fuorviante del ragionamento induttivo e probabilistico. È in questo modo che ci si espone, sempre secondo la terminologia di Taleb, ai "cigni neri", agli eventi inaspettati che non potevano essere previsti perché non was no way to predict them, within the assumptions adopted previously in which the forecasts were made.
Any scientific law is an example of reverse inference: after observing a number of white swans, I can formulate the hypothesis that all swans are white universe, exposing them to risk, however, inevitable and incalculable, the swan black. Inevitable because we can not be sure of the correctness of the assumptions on which we merge, what is the description of the universe that would allow us to make inferences direct and therefore very reliable. One is reminded also of the turkey Russell, convinced that on Christmas Day would bring him food because so had done all the other days of the year.
Taleb, however, despite all the hatred that spreads both hands, in his book , for philosophers and experts in probability for any reason, has not discovered or theorized him first the limits of induction (already explored by Hume, Goodman and others who find it unnecessary to mention). In summary, the problem of inductive-probabilistic assumptions ("almost certainly the next ball to remove the urn will be white") is that all are based, in turn, on assumptions ("urn there are 99 white balls and a black one "), whose reliability of which is subject to the calculation of probabilities, giving start a vicious circle.
vicious circle that can be broken, perhaps noting that the scientific hypotheses, natural laws and theories are by no means simple empirical generalizations that can be put under consideration in the calculation of probabilities, but something more. But that's another story.
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