Copyright AgencyIlnliled (CAL) licensed cop;..l'urth~coaing and Communication.prcfilbircd except onpayment of fee per~op'yor Commuication And otherwise iil~a~;&brd:1nce wnh the licence from CAf:.-fQ /\C~R.For more Information contatd::AL 0" (02)939·1-7600 or [email protected] Analysing Football Finals with a'Spreadsheet Stephen R. Clarke Swinburne University of Technology Using sporting applications to make mathematics more interesting to both secondary and tertiary students is often discussed in the literature, eg Clarke (1984,1991), de Mestre (1987). Australian Rules Football, being the winter sport followed by most people in the southern states of Australia, certainlyqualifies as being of interest to students, and several authors (Clarke (1984,1988),Weal (1987),Schwertman & Howard (1989),Watson (1991)), have suggested ways to incorporate football into the classroom. In this article, we show that a mathematical analysis capable of being performed by secondary students can shed light on the appropriateness of the current AFL finals system - a topic causing much discussion on radio and TV and in the daily press. To generate the 128 equally likelyresults the aid of a spreadsheet (or a word processor) is enlisted. Introduction From 1931 until 1971, the Victorian Football League (VFL) finals were played underthe 'Page final four' system. Teams in third and fourth position played a knockout first semi-final, the loser being eliminated and the winner going into the preliminary final. The top two played a second semi-final- the winner going straight into the grand final and the loser getting a 'second chance' by playing in the preliminary final.. The winner of the preliminary final then progressed to the grandfinal. In 1972 the VFL introduced a final five, played under the 'Mclntyre Final Five' system. By1991, with the introduction of interstate clubs, the competition had grown to 15 teams, and the Australian Football League (AFL) which was by now the governing body, introduced a new finals system played between the top 6 teams. After some criticism, they adjusted the system again for 1992 and introduced the 'McIntyre Final Six' system. This system of matches includes 2 double chance matches and 5 knockout matches to produce an eventual premier. The first week of finals eliminates 2 teams and produces a top four, from where the finals are played exactly as they were under the Page final four system. In week 1, the top two teams play in the qualifying final, the winner retaining its double chance by going into the old position 1, the loser to position 3 (in1992 Geelong beat Footscray). In the other two elimination finals, 3 plays 6 (Collingwood lost to St Kilda) and 4 plays 5 (West Coast beat Hawthorn), the losers being eliminated. The highest ranked winner (West Coast) gains a double chance and goes into position 2, and the other winner (St Kilda) to position 4. The most controversial aspect of the system is that the path the winner of 4 and 5 takes is determined by who wins between 3 and 6. Thus in 1992, because St Kilda beat Collingwood, West Coast went straight into the second semi-final. Had Collingwood won, West Coast would have gone into the first semi-final and had no double chance. In week 2 of the finals in 1992, West Coast beat Geelong to go into the grand final, and Footscray defeated St Kilda. In week 3, Geelong qualified for the grand final by beating Footscray, and in the following week West Coast won the Grand Final. The order of the teams after each week of the finals is shown in table 1. 35 Stephen R. Clarke Table 1 Order of teams after each week of 1992 finals. Week1 Week2 Week 3 Week4 Geelong Geelong West Coast West Coast Footscray West Coast Geelong Geelong Collingwood Footscray Footscray Footscray West Coast St Kilda St Kilda St Kilda Hawthorn Collingwood Collingwood Collingwood St Kilda Hawthorn Hawthorn Hawthorn Schwertmanand Howard (1989, 1990) look at a probability model for the AFL Finals seriesas it was played from 1972until 1990 - a seriesof 6 games betweenthe top 5 teams. They list the 4 paths that result in the 4th team winning the grand final, and the 16 paths that result in the second team winning. Forthe top team they say"Directcomputation of the probability that team A wins the grand final is quite involved, with many different paths"andsuggestindirect methods. The current system,witha final 6playing aseriesof 7 matches has the extra complication that the position of a winningteam now depends not just on their match but on results of other matches. Forthe finalsixsystem,wewish to calculate not onlythe chance of eachteam winning the grand final, but also someother probabilities of interest such as the chance of pairsof teamsmakingthe grand finaland the chance of eachteamfinishing inanyposition. All the probabilities wouldfollow from the chance of allpossible finishing orders. Sothe problem is, for example, given the original order before the finals of Ceelong, Footscray, Collingwood, WestCoast, Hawthorn, St Kilda, what is the probability of a finalfinishing order suchas that whichultimately occurred, of WestCoast,Ceelong,Footscray, StKilda, Collingwood, Hawthorn? Development of all possible outcomes Supposewe designateeachteam bytheir finishing position at the endof the homeand away matches. (In a classroom situation, teachers may wish to make the example more concretebyusingthe actualnamesof the teams,as we have donein table1).One possible sequence of match results (the one that actually occurredin 1992) could be: week 1: 1-2 (1 wins), 3-6 (6 wins), 4-5 (4 wins); this produces an order after week 1 of 142635. week2: 1-4 (4wins), 2-6 (2wins); this produces an order after week 2 of 412635. Preliminary final: 1-2(1 wins); this produces an order after the preliminary final of 412635 Grand final: 4-1 (4winslthlsproduces a final finishing order of 412635. There is no obvious pattern between the finalorder and the match results. Also there are other match results that also produce the same finishing order. For example, team 1 could loseits matchin week1 and the same final order could stillresult. It wouldbequite tediousto workout byhand allpossible 2' =128sequencesof matchresults.The factthat the positions of teams depend not just on the results of their matches but the results of 36 Analysing Football Results with a Spreadsheet others further complicates matters. Howeveraspreadsheet(orawordprocessor) comes to our aid. These packages allow the copying and movementof columns as wellas rows of text. Using this facility, the 128 sequences can begeneratedeasily ina normallessontime. The methodinvolves keeping not just the current order asabove but allpossible orders after eachmatch. Everymatchhas 2 possible results,eachofwhichproduces an associated changein the order. Before the final seriesthe order is 1, 2, 3, 4, 5, 6, so we have Order 1, 2, 3,4 , 5,6 Let us considerthe match between 1 and 2. This can have 2 results, so we copy the wholerow twice. Nowif1 beats2 the orderstaysthe same,soweleave the first rowalone, but if 2 beats 1 they interchange, so we do this to the second row. This gives us I, 2, 3, 4, 5, 6 2, I, 3, 4, 5, 6 Werepeat the process for the match between4 and 5. This works in exactly the same way,with a duplication of the wholetableand aswapof twocolumnsin the second halfof the tableto give I, 2, 3, 4, 5, 6 2, I, 3, 4, 5, 6 1, 2, 3, 5, 4, 6 2, I, 3, 5, 4, 6 The next match, 3 versus6 is the mostcomplicated of the wholeprocedure. Againwe copythe wholeset of4 rows twice. If 3 wins,the positions stay the same,so weleavethe first 4 rows unchanged. If 3 loses, it moves to 5th while6 movesto 4th, so againwe move the relevant columns in the second half of the table. Finally, because the highest ranked winneractually jumps overthe loserofthematchbetween1 and2,weswapthe second and third columnin the table to give I, 3, 2, 4, 5, 6 2, 3, I, 4, 5, 6 I, 31 21 5, 4, 6 2,3, I, 5, 4, 6 1, 41 21 6, 3, 5 2, 41 I, 6, 3, 5 1, 5, 2, 6, 31 4 2, 5, I, 6, 31 4 Thus we have all possible orderings after the first weekendof the finals. It might be argued that the table up to this point could be more easily built up from first principles. Howeverfrom here that method could no longerbe usedas the positions of teamsdepend on previousresults. Forexample there arenow2 possibilities for1st place, 3possibilities for 2nd place, 2 for 3rd and3 for 4th. The remaining 4 matchesare quite straight forward, as 37 Stephen R. Clarke from here on the result of any match either retains or swaps the order of the two teams involved. Eachmatch requires only 2 operations - a complete duplication of the whole table and a move in the second half of the table of the column for the lower placed team. The various stages are shown below - although for brevity we have only shown the first few and last few rows. Note each step doubles the number of rows. First semi-final: current 3 plays current 4 (16 rows) I, 3, 2, 4, 5, 6 2, 3, I, 4, 5, 6 1,3, 2, 5, 4, 6 2, 3, I, 5, 4, 6 I, 4, 6, 2, 3, 5 2,4,6, I, 3/ 5 1, 5, 6, 2, 3,4 2,5,6, I, 3, 4 Second semi-final: current 1 plays current 2 (32 rows) 1,3,2, 4, 5, 6 2,3, 1, 4, 5/ 6 1,3,2,5,4,6 2, 3, I, 5, 4, 6 4, I, 6, 2, 3, 5 4, 2, 6, 1, 3, 5 5/ 1, 6, 2, 3, 4 5, 2, 6, I, 3, 4 Preliminary final: current 2 plays current 3 (64 rows) 1, 3, 2, 4, 5, 6 2, 3/ 1/ 41 5, 6 11 3, 2, 5, 4, 6 2, 3, I, 5, 4, 6 4/ 6, 1,2, 3, 5 4,6, 2, I, 3, 5 5/6, I, 2, 3, 4 5,6,2,1,3,4 Grand final: current 1 plays current 2 (128 rows) 1,3,2,4,5,6 2, 3, 1, 4, 5, 6 1,3,2, 5, 4, 6 2, 3, 1/ 5, 4, 6 6,4, I, 2, 3, 5 6, 4, 2} I, 3, 5 6, 5, I} 2} 3, 4 6, Sf 2, 1, 3, 4 38 Analysing Football Results with a Spreadsheet We now have a list of the 128 possible outcomes, Because of the systematic method used to derive the above there are some interesting patterns.