Probabilistic Paradox

The probabilistic paradoxes are the largely against-intuitive problems of the theory of probability or quite simply having various results according to the interpretation which one or not makes of the statement among several legitimate possibilities (in this last case the word Paradoxe is of course an abuse language).

Simple against-intuitive results

It is the case of the Paradoxe of the birthdays: to believe that the probabilities obey linear laws led to answer 180, whereas the good answer seems ridiculously weak.

Ambiguous statements

The Paradoxe of Bertrand raises a question in which the " randomly " can indicate several different methods of pulling, and leading to contradictory résutats.

Implicit use of a Conditional probability

The paradoxes of this category are relatively simple problems with a rigorous approach but where the intuition leads to abérants results. Some become even subjects with trolls, the anglophone wikipédia having often been used as battle field on this subject.

One can classify them thus, at least with more disputed:

In the case of the three coins, it is easy to do oneself total calculation to note that there is only one chance out of four and not out of two to draw three parts on the same side. On the other hand to correct the fallacious reasoning giving answer 1/2 is more delicate. The paradox of the prisoners seems a handling rather immediately: it is rather visible that a simple matter cannot increase the chances to survive whatever the answer . In the case of the two children, the statement becomes rather ambiguous and one hesitates to employ the conditional probabilities. Especially, the correct reasoning is almost always false in the everyday life. The case of the Monty-Hall is a recurring ground for dispute: the reasoning presented so often jumps of the stages which it is possible to criticize them. The correct result is all the more difficult to understand that it rests in fact on the perverse dealing of a third, the presenter. The problem of Beautiful with wood sleeping rests in fact on the difficulty of knowing which options to distinguish as equiprobable when there are several junctions.

There are similarities between the fallacious reasoning employed: for example in the case of the coins, it is legitimate to say that if two parts present the same side, a third has a chance on two to present the same one. Except if one selected which would be the third part precisely so that the two others check the property. Skew is thus to have forgotten that the choice of the third part was perhaps not made randomly, and that this question changes all.

Same manner, if a family comprises two children introduces a boy, one can think that the other child has a chance on two to be a brother. Except if the introduced child were precisely selected because it is a boy, situation which does not occur that within the framework of this problem. In the same way, in the problems of the prisoners or the Monty-Hall, one forgets that the presenter or the guard was obligatorily to make a revelation corresponding to a case " perdant" (which always exists): nothing changed for the initial door or the prisoner questioner.

Another manner of obtaining false results is to make forget the driving advance with the final situation. From where answer 1/2 in the case of the Monty-Hall or of Beautiful or wood sleeping, which falls if one traces a tree representing the successive events.

It is important to announce that the solutions having finished by imposing itself are not based on the argument of authority but on the checking of the arguments suggested: if the bad spontaneous answers of the majority of people seems to place the problems at very an high level, a Arbre of probability is generally enough to reveal the good solution.

The other difficulties are related to the difficulty of perceiving the probabilities like referring to information and not to physical reality, and that to evaluate the subjective Probabilité in a given situation.

Jean-Paul Delahaye estimates that the Argument of the apocalypse also belongs to this category.

Popularity

The described problems above made the object of baited enough debates, and the neophytes as the professional mathematicians were often traps by one of these problem, or by alternatives (especially, it is easy to propose alternatives such as one supposes the identical result intuitively whereas it is different). Especially, the problems of the Monty-Hall and Beautiful with wood sleeping are the subject still of debates baited on Internet (in spite of the experimental confirmation for the Monty-Hall), so much so that the neologisms " halfer" (" demiste") and " thirder" (tierist) were invented to designate the defenders of each answer.

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