Un singe aurait pu écrire Harry Potter !

Soyons clair : le titre de cet article ne contient aucune insinuation sur la qualité d'écriture de J.K. Rowling - l'auteur de Harry Potter - mais nous invite au contraire à nous pencher un peu sur la science des grands nombres et sur les calculs de probabilités. Pas de panique, vous allez voir, ce n'est pas si dur !
 
Imaginez que l'on puisse faire choisir à notre singe une lettre parmi les 26 de l'alphabet, en faisant en sorte qu'il ai autant de chance de prendre le A que le Z (par exemple en piochant les yeux fermés dans un sac contenant une fois chaque lettre). On dira alors que le tirage est équiprobable. Comme au loto, il faut en effet garantir que chaque lettre ai autant de chance d'être tirée qu'un autre.
 
Supposons alors que notre singe dispose d'une machine à écrire permettant uniquement de faire des lettres minuscules par pression sur une touche. Cette machine a donc les lettres de a à z, soit 26 caractères. Sachant qu'il ne comprend pas le language humain, si vous lui demandez de taper banane, notre singe a un chance sur 26 de taper au hasard sur la touche b ( soit 4%). Ensuite il a encore une chance sur 26 de taper a. La probabilité de taper consécutivement ces deux premières lettres "ba" est donc 1 chance sur 26x26 = une chance sur 676. Pour obtenir le mot complet "banane" (de six lettres), il aura donc une chance sur 26x26x26x26x26x26 = 266, soit une chance sur 308 915 776.
Le tirage suivant, de 15 lettres, contient par exemple le mot banane : lsdhgbananeldfg. Je ne ferais pas la démonstration mathématique ici, mais on considère qu'un tirage de n lettres est presque sûr de contenir le mot banane, et ce d'autant plus que n est un grand nombre. En fait, si n est énorme, voir quasi-infini, alors il y a presque 100% de chances que le tirage contienne le texte intégrale d'Hamlet, de Harry Potter, ainsi que votre biographie, la mienne et même tous les livres et toutes les histoires de la terre. Et ce "presque" et ce "quasi" sont même des nombres que l'on peut calculer !
 
Logo astuce
Quelle est la chance de trouver la combinaison d'un cadenas à 4 chiffres ?
Les chiffres d'un cadenas vont de 0 à 9, il y a donc 10 possibilités pour chaque roues du cadenas. Cela donne une chance sur 10 pour la première roue, une chance sur 10 pour la deuxième, et ainsi que suite, soit une chance sur 10x10x10x10 = une chance sur 10000 au total de trouver la combinaison. En supposant que par malchance votre code soit le 9999 et que le voleur commence par 0000, 0001, 0002, etc.. au rythme d'un essai toutes les deux secondes, il lui faudra au maximum environ 5h30 pour trouver votre code - et beaucoup de patience !
 
Mais revenons à notre singe et remplaçons-le par un ordinateur qui ira bien plus vite. Considérons aussi Hamlet dans une version sans accents, sans majuscules, sans ponctuation et sans espaces entre les mots ; bref, un hamlet écrit uniquement avec nos 26 lettres. Pour écrire les 20 premiers caractères du livre dans le bon ordre,  il faudra à notre ordinateur une chance sur 2620, soit une chance sur 19 928 148 895 209 409 152 340 197 376. Vous avez à peu près autant de chance de gagner 4 fois de suite au loto !
Alors pour écrire Hamlet en entier - environ 35 400 mots, admettons 180000 lettres -, l'ordinateur a une chance sur 26180000 de produire le texte entier, soit 0, 000...cent quatre-vingt mille zéros et 1 pour cent de chances d'y arriver !!
 
Vous ne vous rendez pas compte à quel point ces chiffres sont énorme ? Celà veut dire que statisquement parlant, il faudra que notre ordinateur produise 26180000 lettres aléatoirement pour que parmi elles se cachent le texte intégral d'Hamlet. Si cet ordinateur pouvait tirer une lettre toutes les nanosecondes (il y a en 1 000 000 000 dans une seconde), il faudrait malgré tout plusieurs fois l'âge de l'univers pour y arriver....
 
Possible en théorie, impossible en pratique !
Votre note : Aucun(e) Moyenne : 4 (5 votes)

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