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Generalization through problem solving

Generalization through problem solving. Part III .-IV . Probability through statistics. Gergely Wintsche Mathematics Teaching and Didactic Center Faculty of Science Eötvös Loránd University, Budapest.

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Generalization through problem solving

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  1. Generalization throughproblemsolving Part III.-IV. Probabilitythroughstatistics Gergely Wintsche Mathematics Teaching and Didactic Center Faculty of Science Eötvös Loránd University, Budapest CME12, 2012.07.04.– Rzeszów, Poland Gergely Wintsche

  2. Outline 1. Back tothefuture 2. Experiments 3. Head runs 4. Orderingtheprobabilities Part III / 2 – Probability, experiments, statistic Gergely Wintsche

  3. Back tothefuture Tomatoes Originallyit is comingfromthegenerations of pea. 1. Whatdoes a monogenicinheritancemean? 2. Whatarerecessive-dominantalleles? 3. Whatdoeshomozygous, heterozygousmean? Part III / 3 – Probability, experiments, statistic Gergely Wintsche

  4. Inheritance Tomatoes Wehaveplentyof tomatoesyellow and redones. Firstgeneration red × red red × red red× yellow yellow × yellow red × yellow Secondgeneration 61 red 47 red, 16 yellow 58 red 64 yellow 33 red, 36 yellow • Whiccolorwasthedominantfenotype? • Whatarethepossible and whataretheprobablefenotypesintheabovecases? Part III / 4 – Probability, experiments, statistic Gergely Wintsche

  5. Experiments Die Therearethreegivenexperiments. The player must chooseone of them and repeatit 20 times. He writes down onlytheresults of theexperiment. The othershavetofind out whatwasthechoice of theplayer. A: Onethrowsa die and writes 0 iftheresultwas 1 or 2, writes 1 iftheresultwas 3 or 4 and writes 2 iftheresultwas 5 or 6. B: Onethrowsa die and writes 0 iftheresultwas1, 2 or3, writes 1 iftheresultwas4 or5 and writes 2 iftheresultwas6. C: Onethrowsa die and writes 0 iftheresultwas1, writes 1 iftheresultwas2, 3, 4 or 5 and writes 2 iftheresultwas 6. Part III / 5 – Probability, experiments, statistic Gergely Wintsche

  6. Experiments Die Afterthefirstexperimentwegotthe series: 2, 2, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 0, 0 Trytofind out theplayer’s choice! Howcouldyoumake a decision? Doyoumakethecorrectdecisionafterallresults? Part III / 6 – Probability, experiments, statistic Gergely Wintsche

  7. Head runs Click here to play movie Head orTails Youget a paper. Therearetwoemptytablesonitsignedwith A and B. Both tableshave 100 fields. Chooseone of thetable and write H or T inthefieldsasyoucanimagine a random headortail series. Youdonotnecessarilyhavetochoosetable A. Pleasereadthepagefullbeforeyou start it. Take a coinrunthetrial and fillintheothertablewiththerealdata series. Shareyourtableswithme and withyourneighboursandtrytofind out whichtablewasartificiallycompleted and which series originatedfromtherealcointossingexperiment. Discussyourdecisionmethod. Part III / 7 – Probability, experiments, statistic Gergely Wintsche

  8. Head runs Head runs List allpossiblesequencesin a threelong H-T series. Derivetheprobability of thelongestheadrun. Letussupposethatthecoin is fair, sotheprobability of head and tail is ½. Calculatetheaveragelength of thelongestheadrun. 1/8 · 0 + 4/8 · 1 + 2/8 · 2 + 1/8 ·3 = 11/8 Part III / 8 – Probability, experiments, statistic Gergely Wintsche

  9. Head runs Head runs Couldyourepeattheabovecalculationsfor 4, 5, 6, …, n longcointossing series? Part III / 9 – Probability, experiments, statistic Gergely Wintsche

  10. Head runs Head run I saythatitwill be easiertodealwiththecumulativenumbers. An(x):= The number of sequencesoflengthninwhichthelongestrun of heads is at most x. Part III / 10 – Probability, experiments, statistic Gergely Wintsche

  11. Head runs Head runs Letusseethek = 3 case more accurately. The trivialquestion: If n = 1, 2 or 3 thenhowmanyoutcomesarefavourable? Ifn ≤ 3 thenAn(3) = 2nclearly (anyoutcome is OK). Whatcould be thebeginning of a series ifn > 3. Ifn > 3 theneachfavorablesequence must beginwithT, HT, HHTorHHHT and it is followedby a subsequencehaving no more thanthreeconseqcutiveheads. Part III / 11 – Probability, experiments, statistic Gergely Wintsche

  12. Head runs Head runs Part III / 12 – Probability, experiments, statistic Gergely Wintsche

  13. Head runs Head runs Couldyouwrite a recursion formula forAn(3)? The properrecursion is An(3) =An -1(3) + An -2(3) + An -3(3) + An -4(3) forn > 3 The values of An(3) are Part III / 13 – Probability, experiments, statistic Gergely Wintsche

  14. Head runs Headsortails Wecancode a TH sequenceinto a onesignshorterDSsequence. IfwedenoteBn(x) thenumber of sequenceswhenthelongest H or T run is at most x, then Bn(x) = 2An – 1(x – 1) for x ≥ 1 Part III / 14 – Probability, experiments, statistic Gergely Wintsche

  15. Head runs Head ortails / average Let P(head) = p and P(tail) = q. (p + q = 1) Ifit is a fair cointhenp = q = ½. Everyheadrunstartsatthebeginningorafter a tail. (Wewillallow 0 longheadruns.) Therewill be ≈nqheadruns. Therewill be ≈nqpheadrunscontainsatleastonehead. Therewill be ≈nqp2headrunscontainsatleasttwoheads. Therewill be ≈nqpxheadrunscontainsatleastx heads. Ifp = ½ then If Part III / 15 – Probability, experiments, statistic Gergely Wintsche

  16. Head runs Whichdice is better Andrew has threedice. Theyarenotregularbutthepointsonthem: X: 3, 3, 3, 3, 3, 6 Y: 2, 2, 2, 5, 5, 5 Z: 1, 4, 4, 4, 4, 4 Bruno thinksit over verycarefully and choosesone of thedice. Afterthis Andrew choosesone of thetworemainderdice. Both of themthrowtheirown die and thebigger is thewinner. Is it a fair game ornot? Who has a chancetowin? If Part III / 16 – Probability, experiments, statistic Gergely Wintsche

  17. Head runs Which series arebetter Andrew and Bruno play a new game. They flip a coinseveraltimes and iftherearetwoconsecutiveheadsbefore a head-tailpairinthisorderthen Andrew winotherwise Bruno. Forexample: TTHH (Andrew win), TTHT (Bruno win) Is it a fair game? If Part III / 17 – Probability, experiments, statistic Gergely Wintsche

  18. Head runs Which series arebetter Andrew and Bruno play a new game. They flip a coinseveraltimes and iftherearetwoconsecutiveheadsbefore a head-tailpairinthisorderthen Andrew winotherwise Bruno. Forexample: TTHH (Andrew win), TTHT (Bruno win) Is it a fair game? START 1/2 1/2 H T 1/2 1/2 H T If Part III / 18 – Probability, experiments, statistic Gergely Wintsche

  19. Head runs Which series arebetter Andrew suggested a bit longer series. He is readytochoosetheHHT and Bruno theHTTtriples. They flip a coinseveraltimes and those boy winwhosetriplecomesearlier. Is it a fair game ornot? If Part III / 19 – Probability, experiments, statistic Gergely Wintsche

  20. Head runs Which series arebetter P2 Pidenotesthat Andrew is here and win. P1 T HT HTT START H HH HHT P3 P4 If Part III / 20 – Probability, experiments, statistic Gergely Wintsche

  21. Head runs Which series arebetter Letusrepeatthecalculationforthe HHTHTT (TTH) THHHHT inpairs. If Part III / 21 – Probability, experiments, statistic Gergely Wintsche

  22. Head runs Which series arebetter Andrew and Bruno throwtworegulardice. Andrew winifthe sum of them is 12. Bruno winiftwoconsecutivethrowsare 7. What is theprobabilitythat Andrew winthe game? Drawthegraph of the game. 12 12 29/36 1/36 START 1/36 29/36 7 7 6/36 6/36 If Part III / 22 – Probability, experiments, statistic Gergely Wintsche

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