Dice and Data A comic about statistics with a 95% chance of being one Dice and Data A comic about statistics with a 95% chance of being one GOVERN DE LES ILLES BALEARS Vicepresidència i Conselleria d’Economia, Comerç i Indústria Direcció General d’Economia CCIX © Edition: Directorate General of Economy Project Management: Antoni Monserrat I Moll. General Director of Economy General Coordination: Jose Antonio Pipo Jaldo. Director of IBAE Produced by: Balearic Institute of Statistics Palau Reial, 19 07001 – Palma (Mallorca) - Spain Telephone: (34) 971-176-706 http://ibae.caib.es E-mail: [email protected] Author: Javier Cubero Management and Production: Inrevés SLL Illustrations: Alex Fito and Linhart Color: Pau Genestra Layout: Xisco Alario and Margalida Capó Script adapted by: Felipe Hernández Translation to English: T&i-Traducción e interpretación Revision: Roser Belmonte Juan Coordination: Sebastià Marí and Pere Joan Collection: Estadística al carrer. Volume 1 Title: Dice and Data. A comic about statistics with a 95% chance of being one No. IBAE: CCIX Legal Deposit: 2PM-2462-2002 ISBN: 84-89745-53-6 Printed: Planografica Balear SL Publishing date of the English version: November 2002 © Copyright: Directorate General of Economy. Regional Ministry of Economy, Trade and Industry. Balearic Islands PRESENTATION The study of mathematics and the concepts of statistics have always been known to be difficult disciplines that are not very attractive to students. Therefore, the Government of the Balearic Islands wanted to contribute to the Worldwide Year of Mathematics in the area of the popularization of these areas with the publishing of the comic Dice and Data. The present edition, elaborated by the Balearic Institute of Statistics (IBAE) of the Regional Ministry of Economy, Trade, and Industry, is an effective instrument that meets the teaching requirements of the Spanish High School study plan (ESO) and can be used to educate adults. This document’s purpose is to enable the aforementioned people to learn the information it contains in an easier and more attractive way. This publication is included in the teaching plan that was initiated by the IBAE with the purpose of communicating to the society the different studies and analyses that have been carried out by the IBAE. However, the objective of the institute is not only to make known this statistical data that shows the socio-economic reality of the Balearic Islands, but also to present a work methodology that is important in planning decisions regarding the future of our country, building on a good solid base. Finally, we would like to thank the team that worked on this publication and their par- ticipation in an experience that we consider to be innovative. We would also like to make special mention of the group of creative and graphic designers that participated in the making of this publication, having demonstrated the high quality of this sector in the Balearic Islands. Pere Sampol i Mas Vice-president of the Government of the Balearic Islands and Regional Ministry of Economy, Trade and Industry PROLOGUE It is a true pleasure to preface the work that you presently have in your hands for many reasons. First of all, undoubtedly not the most important reason, but nevertheless important, would be the longstanding friendship that I have with the author, since just 35 years ago we were in the same class in the University. In those days, we would have never thought that we would meet up again for the love of statistics. The second is the very work itself, Dice and Data that, as a sharp reader, you will observe that it is much more than a comic. From the choice of names of the characters, that is definitely not random but has its story, like 55’s birthday at the end or the rock group called quartile. To point out some parts that I particularly enjoyed and that will make you think, I will begin with the historical brushstrokes that each chapter begins with, followed by an ele- gant way of explaining the difference between a continuous variable (snail trails) and a discrete variable (grasshopper leaps), and finally the method of teaching that the data contains more information than it appears to contain at first (the age problem of the four siblings) is, in the least, original. The way in which it avoids reasoning by way of charts, since they can lead you to evi- dent errors (areas of the square and rectangle), reminded me of a professor that we once had in our studies of Mathematical Science. The introduction of the concepts of population density and the age profile are very illustrative with their applications to the different cities of the Balearic Islands along with the motor vehicle accident rates, which is something that effects our youth today and that show why there are irregularities in the age profile. It is probably in chapter 8 that the author shows his genius with cartoons used to introduce index numbers depending on the old (unlabeled corked bottles) and new quan- tities (tetra briks of milk), in conjunction with the old prices (spiral notebook) and the new prices (computer screen). May these last words encourage Javier to continue with the work at hand and to delight us in the near future with a sequel. Granada, April 2000 Rafael Herrerías Pleguezuelo Professor of Applied Economics INDEX Chapter 1 – PIERRE DE FERMAT pg. 8 Chapter 2 – THOMAS BAYES pg. 13 Chapter 3 – BLAISE PASCAL pg. 20 Chapter 4 – ADOLPHE QUETELET pg. 26 Chapter 5 – JAKOB BERNOUILLI pg. 34 Chapter 6 – CHARLES DODGSON pg. 43 Chapter 7 – WILLIAM SEALEY GOSSET pg. 50 Chapter 8 – LASPEYRES AND PAASCHE pg. 59 Chapter 9 – DE MOIVRE AND GAUSS pg. 68 ANNEXES pg. 87 THE SUPERHERO The Superhero …THE MOST IMPORTANT CHARACTER IN THIS COMIC: YOU! 6 THE CHARACTERS 55 Riddle Chance Binomial GAUSS Graph 7 CHAPTER 1 PIERRE DE FERMAT French mathematician (1601-1665) His knowledge earned him the nickname “prince of amateurs” He was one of the pioneers of the theory of probabilities. Chapter 1 – PIERRE DE FERMAT THE GRASSHOPPER’S GONNA WIN! WHAT A RACE! I’M GOING TO MAKE A HYPOTHE- OF COURSE THE GRASSHOPPER IS LUCKILY FOR THE SIS: IT’S MOST LIKELY THAT THE GOING TO WIN, HE’S FASTER. SNAIL, THE SNAIL WILL LOSE. GRASSHOPPER STOPS BETWEEN JUMPS. IT’S AS CLEAR AS DAY! THE GIVE ME A PENCIL AND SOME WAIT A MINUTE. THE GRASSHOPPER’S GONNA WIN. PAPER AND I WILL KEEP TRACK GRASSHOPPER ADVANCES BY OF HOW THE RACE IS GOING. JUMPING AND LEAVES TRACKS… …IN CERTAIN PLACES ALONG THE WAY. THIS WAY WE CAN TRACE THE MEANWHILE, THE SNAIL ADVANCES PATH OF THE COMPETITORS. LEAVING A CONTINUOUS TRAIL BEHIND HIM. 9 Chapter 1 – PIERRE DE FERMAT YEAH, BUT THERE’S ONE PROBLEM. WE’LL CALL HIM HA, HA, HA! …WE’LL CALL “MR. SEE HOW THE GRASSHOPPER “MR. DISCRETE”! THEN THE SNAIL… CONTINUOUS”. THINKS BEFORE HE LEAPS… THEY ARE CALLED EVENTS THAT REMINDS ME THAT PERHAPS THE STUDY OF BECAUSE THEY HAPPEN, NOT THEY STUDY EVENTS WITH STATISTICS BEGAN WITH BECAUSE THEY ARE DISAS- “DISCRETE” VARIABLES AND THE STUDY OF GAMES… TROUS, BLOCKHEAD! “CONTINUOUS” VARIABLES. WE COULD MAKE A LAB AND DO SOME EXPERIMENTS! I DON’T KNOW IF THIS HAS GREAT! ANYTHING TO DO WITH IT COOL! BUT I HEARD THAT IT IS HARDER TO GUESS WHAT ONE PERSON WOULD WISH FOR THAN WHAT A MILLION PEOPLE WOULD WISH FOR. THAT JUST GAVE ME AN IDEA! LISTEN. I SAY WE PLAY A GAME… Experiments EXERCISE WE FLIP A COIN IN THE AIR 8 TIMES AND WRITE DOWN THE RESULTS. WE DO IT THREE TIMES AND WRITE DOWN ALL THE TIMES THAT HEADS COMES UP. THEN WE WILL FLIP A COIN 50 TIMES IN A ROW. I BET YOU THAT THE NUMBER OF TIMES THAT THE COIN COMES UP HEADS THIS TIME IS CLOSER TO 25 THAN IT WAS TO 4 WHEN WE FLIPPED IT THE OTHER TIMES. SO WHAT YOUR SAYING IS THAT THE MORE WE REPEAT THE EXPERIMENT, THE MORE SURE WE WILL BE TO GET IT TO COME UP HEADS HALF THE TIME WE FLIP IT. IF WE DID IT A MILLION TIMES, WE WOULD GET IT EVEN CLOSER TO BEING HALF OF THE TIMES. AND IF WE DID IT 10 MILLION TIMES, IT’D BE EVEN CLOSER. THEREFORE, THE PROBABILITY THE COIN WILL COME 1 UP HEADS OR TAILS IS CLOSER TO 2 THE MORE TIMES WE FLIP IT. 10 Chapter 1 – PIERRE DE FERMAT Experiments OK, BUT THAT’S EASY BECAUSE A COIN ONLY HAS TWO SIDES: HEAD AND TAILS, YOU WIN OR YOU LOSE. IF WE TRY THE EXPERIMENT OUT WITH A DIE, WE’LL SEE THAT THERE ARE 6 POSSIBILITIES. LET’S SEE HOW MANY TIMES 5 COMES UP. WE WILL ROLL THE DIE 90 TIMES AND WRITE DOWN THE RESULTS. LET’S SEE HOW IT TURNS OUT. SO, IF WE HAVEN’T CHEATED AND THE DIE IS NOT RIGGED, WE CAN SAY THAT WE HAVE WON 15 TIMES AND HAVE LOST 75 TIMES. 15 1 THE PROBABILITY THAT WE ROLL A 5 IS 90 OR 6 , WHILE THE PROBABILITY OF NOT 75 5 ROLLING A 5 IS 90 OR 6 . SO WE CAN SEE THAT THE PROBABILITY OF NOT ROLLING A 5 IS EQUAL TO 1 MINUS THE PROBABILITY OF SUCCEEDING. SO, THE PROBABILITY OF GUESSING THE OUTCOME OF A SOCCER MATCH BETWEEN TWO EVEN TEAMS WOULD BE 1 (SINCE THERE ARE ONLY THREE POSSIBLE OUTCOMES: 3 2 WIN, LOSS, OR TIE) AND THE PROBABILITY OF NOT GUESSING WOULD BE 3 . WHICH BRINGS US TO SEEING THAT GUESSING IS NOT ALWAYS EASY, JUST AS THE QUESTIONS IN CLASS AREN’T ALWAYS EASY, UNLESS I AM THE ONE ANSWERING.