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Estimation du nombre de citations de papillotes et de blagues Carambar

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Comment évaluer le nombre total de blagues Carambar (ou de citations dans les papillotes Révillon) d'après un échantillon...

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Estimation du nombre de citations de papillotes et de blagues Carambar

  1. 1. Quart d'heure académique du SéminDoc 06/05/2009 LIRMM – Montpellier Estimation du nombre de citations de papillotes et de blagues Carambar Philippe Gambette (équipes MAB/AlGco)
  2. 2. Introduction • Papillotes créées en 1790 - un billet doux pour enrober un chocolat, à l'origine - depuis, rébus, dessins d'humour, citations papillotesrevillon.fr • Carambars créés en 1954 http://fr.wikipedia.org/wiki/Carambar - mélange accidentel de caramel et cacao - devinettes et blagues sur l'emballage depuis 1969
  3. 3. Problématique Combien de citations ou blagues différentes ? • pour le fabricant : - limiter les coûts de production → nombre fini - satisfaire le consommateur • pour le consommateur : - frustration de retomber sur une blague déjà lue - souci d'exhaustivité : combien en manger pour les lire toutes ? • pour le statisticien : - estimer ce nombre n d'après un échantillon
  4. 4. Echantillonnage • tirer un échantillon aléatoire de k papillotes on suppose que les citations sont uniformément réparties dans les sachets • discrétiser les données associer une citation à chaque papillote • identifier les doublons choix de la citation la plus proche du centre du papier associer un entier unique à chaque citation Modélisation de l'échantillonnage : tirer un mot aléatoire de k lettres, choisies parmi un alphabet de n lettres.
  5. 5. Modélisation du problème Sachant qu'il y a un alphabet de n lettres n papillotes différentes au total quelle est la probabilité d'avoir 40 lettres de tirer 40 citations différentes, exactement, dans un mot de 52 lettres parmi un échantillon de 52 papillotes ? Modélisation du problème : trouver la valeur de n qui maximise cette probabilité
  6. 6. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres.
  7. 7. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. nombre de mots de k lettres dont d différentes Pd,k(n) = nombre de mots de k lettres
  8. 8. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. nombre de mots de k lettres dont d différentes Pd,k(n) = nombre de mots de k lettres Nombre ad,k(n) de mots de k lettres dont d différentes : n=3, k=3, d=2 : aab aba abb baa bab bba aac aca acc caa cac cca bbc bcb bcc cbb cbc ccb
  9. 9. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. nombre de mots de k lettres dont d différentes Pd,k(n) = nombre de mots de k lettres Nombre ad,k(n) de mots de k lettres dont d différentes : n=3, k=3, d=2 : aab aba abb baa bab bba on trouve les mots sur d=2 lettres aac aca acc caa cac cca on en déduit les mots sur n lettres bbc bcb bcc cbb cbc ccb par projection.
  10. 10. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. nombre de mots de k lettres dont d différentes Pd,k(n) = nombre de mots de k lettres Nombre ad,k(n) de mots de k lettres dont d différentes : n=3, k=3, d=2 : aab aba abb baa bab bba on trouve les mots sur d=2 lettres aac aca acc caa cac cca on en déduit les mots sur n lettres par projection : ad,k(n) = ad,k(k) Cnd bbc bcb bcc cbb cbc ccb
  11. 11. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. nombre de mots de k lettres dont d différentes Pd,k(n) = nombre de mots de k lettres nk Nombre ad,k(n) de mots de k lettres dont d différentes : ad,k(n) = ad,k(k) Cnd
  12. 12. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. ad,k(k) Cnd Pd,k(n) = nk
  13. 13. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. constante par rapport à n ad,k(k) Cnd Pd,k(n) = nk
  14. 14. Calculs Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. Cnd argmaxn Pd,k(n) = argmaxn nk
  15. 15. Résultats Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. Cnd argmaxn Pd,k(n) = argmaxn nk Pour les papillotes Révillon “Festives” pour k=52 et d=40 : n=93 ? P (n) d,k n 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160
  16. 16. Résultats Trouver la valeur de n qui maximise la probabilité Pd,k(n) de tirer un mot de k lettres ayant exactement d lettres différentes dans un alphabet de n lettres. Cnd argmaxn Pd,k(n) = argmaxn nk Pour les papillotes Révillon “Festives” pour k=52 et d=40 : n=93 ? En fait, n=108, Pd,k(n) soit 14% d'erreur. n 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160
  17. 17. Résultats Evolution du nombre de blagues Carambar “Caramel” estimé en fonction de la taille du tirage : 45 40 valeur de n estimée 35 30 nombre d de blagues 25 différentes trouvées 20 15 10 5 0 nombre k de 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 carambars ouverts
  18. 18. Perspectives • étude de la précision de la méthode par simulations • formule directe pour la valeur de n estimée • utilisations d'autres caractéristiques du tirage pour une évaluation plus précise : - nombre de citations présentes deux fois - distribution des nombres d'apparition de citations - taille la plus longue d'une séquence de blagues consécutives • estimation plus précise du nombre de blagues Carambar • estimation du nombre de surprises Kinder ebay.fr Bientôt sur http://gambette.blogspot.com

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