149 Finals time! AFL
September brings the finals of the Australian Football League. Use your students' interest in this popular sporting code to explore the probabilities of which team from the final eight is more likely to win the Grand Final.
The accompanying software is essential. It allows students to run multiple trials to relate the empirical probability of the simulation to the theoretical probability. Students can also refine the mathematical model by experimenting with using different weightings.
- Lesson notes (pdf includes the lesson, pedagogical pointers, information on the software, and practical hints)
- Final eight board for each group (supplied as a pdf)
- Small sticky notes, one for each student
- Pack of playing cards, divided into four suits from the Ace to the 8
- Dice, two for each group
- Finals time! AFL software:
- Links to the Australian Curriculum
Begin a class discussion about the different sports that students play.
- Is there a 'final' at the end of the season?
- Is the final series as fair as possible? How do you know?
Explore three scenarios: two top teams, three top teams and four top teams.
Two top teams
The two top teams qualify and play off in the Grand Final.
Three top teams: A, B and C
Team A goes straight into the Grand Final.
Teams B and C play off with the winner going into the Grand Final against Team A.
Discuss why this is fair.
As a reward for finishing on top of the ladder, Team A gets into the Grand Final directly.
Four top teams: A, B, C and D
There are two main options.
Team A plays Team D, and Team B plays Team C.
The two winning teams play each other in the Grand Final.
Discuss why this is fair.
As a reward for finishing at the top of the ladder, Team A plays Team D, presumably the weakest of the four.
Team A plays Team B, the winner going straight into the Grand Final.
The loser of that match plays the winner of the match between Team C and Team D to see which team goes into the Grand Final.
Discuss why this is fair.
Teams A and B have a 'double chance' as reward for finishing at the top.
The Australian Football League uses a final eight system. Print out the game board(s) on A3 paper. Gather the class around a central spot and select eight students to demonstrate how the system works.
Demonstration of the final eight system
The eight players represent the eight teams which have qualified for the finals. Ask each student to write his or her name on a small sticky note. At this point, it is best not to use the names of football teams.
Deal out the eight cards to the players. This randomly allocates their positions at the end of the home and away season. Each player places their name on the game board in the correct position.
Play each match from the 'top down'. Assume each match is a 50:50 chance.
Both players roll one dice each and the higher number wins. If there is a draw, there is a re-match (i.e. both roll again). It creates more interest if one player rolls and then the other follows, rather than rolling the two dice at the same time.
As each game is played, players move their name cards according the results.
Play out the games until the Grand Final is reached and a winner is decided. Students should easily see that the top four teams have a double chance.
Play some finals
Following the demonstration, organise each group of eight to play about five final series.
After each final series, the eight cards are dealt again so in the next game every player has the chance to start in a different position.
As soon as there is a winner, that person goes to the board and records a tally mark against the position that they started from.
What are the chances?
Although there are usually only about 15 to 20 data tally marks, the results should clearly show the structural advantage of the double chance given to the top four teams. Ask the class to draw some conclusions. There is no need at this stage to formally calculate the probabilities. The software will assist in this later.
So, while it is called a 'double chance' it actually gives the top four teams three times the chance of the bottom four. And given that the teams are unlikely to be equally matched, the advantage is actually greater. Of course, the empirical class data may not produce the theoretical outcomes.
Which finals combinations are possible?
What is the greatest number of combinations of teams that could play in the Grand Final? Could Teams 1 and 4 meet? What about Teams 5 and 8?
Record all the combinations of any two teams. You may choose to do this as a class or in the team groups. Which combinations will lead to a possible Grand Final meeting? These are good exercises in logically recording possibilities.
There are 28 combinations.
Commonly, students judge the combination of Teams 1 and 6 as impossible, as they could both meet in the first Preliminary Final. However, if Team 1 loses the first game, it is indeed possible. Twenty-six of the twenty-eight possibilities can occur; only 5 & 8 and 6 & 7 are not possible as these teams play in the accurately named Elimination Finals.
Students usually appreciate the logical cleverness of the structure, particularly the cross-over after the Semi-finals, which guarantees that it is not possible for any two teams to play each other twice in the first three weeks. There are other advantages built into the system such as winners hosting in their home state.
Further exploration using the software
Option 1 and 2
This replicates the simulation with the dice.
When students are using the software for the first time, it is preferable to use student names rather than AFL team names, to keep the focus on structure rather than 'barracking' for a particular team.
Students can type their names in any order, and then click and drag the names into the desired positions. Then click <OK>.
Alternatively, after entering their names, clicking on <Random Pick> will randomly allocate an order. When happy with the result, click <OK>.
Play the finals series by clicking on <Play Finals>.
Play many trials. The maximum number of trials is 16 000 to highlight the probability proportions.
This example gives the results from 16 000 trials and clearly shows that the top four teams have approximately triple the chance of winning the Premiership.
Wrap it up
Students should observe that the top four teams have a triple chance (3 in 16 or 18.75%), not a double chance. The bottom four teams have a chance of 1 in 16 or 6.25%.
Using formal calculations, and assuming an equal probability of each team winning, Teams 5 and 8 would have to win four games in a row to win the Grand Final. The chance of this is (1/2)4 or 1 in 16. Again, assuming an equal probability of each team winning, Teams 1 and 4 have the same chance of winning the Grand Final. They either win three games in a row, skipping Week 2, with a probability of (1/2)3 or 1 in 8, or lose the first game and then win the next three with a probability of (1/2)4 or 1 in 16. This results in an overall probability of 1/8 plus 1/16, therefore 3 in 16.
Option 4 of the software
Mathematical models always try to make their assumptions close to reality. The initial mathematical model assumes that each team has a 50:50 chance of winning, which is clearly not true, as the higher ranked teams are 'better'.
One improvement on the model could be that in any game where both players roll the same number, the higher ranked team would be the winner. Students could be invited to invent some rules for the game so it more closely reflected reality.
Improving the mathematical model
Option 4 of the software attempts to make the simulation model slightly more realistic by allowing a selected percentage advantage to be given to any team which has a higher position. For example, by selecting a 1% advantage, then Team 1 would have a 1% advantage over Team 2. If Team 1 played Team 3 they would have a 2% advantage, etc. Similarly, if Team 5 played Team 8, they would have a 3% advantage.
If 14% is selected and Team 1 plays Team 8, they will enjoy a 98% advantage (14% x 7 positions).
Invite students to experiment with different percentage advantages and to notice the patterns of results. If 1000 trials or 10 000 trials are selected, then the results table can easily be read in percentage terms. Ask them to justify their percentage selection with respect to factors other than the position in the final eight (e.g. home ground advantage, weather conditions).
- How soon does the percentage of wins figure become stable? Why might this be?
- What percentage advantage allows team 1 to win 30% of the finals?
- What percentage advantage allows Team 8 to win once every 100 years (i.e. a 1% chance)?
- Is the structural 3 to 1 advantage of Team 4 over Team 5 maintained for various percentage advantages?
Laminating the game boards is a good idea.
Students may wish to have a copy of the Final eight board to share with family and friends.
Twenty-four students in the class is ideal for three groups of eight. Extra students can work together to represent one of the eight teams on the board.