Arsenal collection: who's the winner?
When two closely matched teams play each other, what is the most likely result?
Age
11 to 14
Challenge level
Problem
Image
When the Arsenal women's team plays Chelsea Ladies, they are equally matched - at any point in the match, either team is equally likely to score.
What are the possible results if 2 goals are scored in total?
Why are they not all equally likely?
This mathematical model assumes that when a goal is scored, the probabilities do not change. Is this a reasonable assumption?
Alison suggests that after a team scores, they are then twice as likely to score the next goal as well, because they are feeling more confident.
What are the probabilities of each result according to Alison's model?
Charlie thinks that after a team scores, the opposing team are twice as likely to score the next goal, because they start trying harder.
What are the probabilities of each result according to Charlie's model?
The models could apply to any football teams or even other team sports where a small number of goals are typically scored.
You could find some data for matches between closely matched teams that finished with two goals and see which model fits most closely to what happened.
You will need to make some assumptions about what it means for teams to be "closely matched". Can you explain the reasoning behind the assumptions you chose to make?
Getting Started
You could simulate the goal scoring using dice - an odd number
could represent a goal for the teachers and an even number could
represent a goal for the students.
Simulate a few games where two goals are scored and record how many "wins", "draws" and "losses" occur.
Simulate a few games where two goals are scored and record how many "wins", "draws" and "losses" occur.
Student Solutions
Thank you to Anna from Mt Eliza North Primary School, Australia, who sent us this solution (originally to another version of this problem):
1. What are the possible results if 2 goals are scored in total?
2-0, 0-2, 1-1.
2. Why are they not all equally likely?
They are not all equally likely because there are two ways you can get the end result, 1-1, by [Arsenal] scoring first and then [Chelsea] or [Chelsea] scoring first and then [Arsenal]. There is one way for the end result, 2-0, [Arsenal] gets the 2 goals without [Chelsea] getting 1. And there is only one way you can get the end result 0-2, by [Chelsea] getting both the goals. Therefore the end result 1-1 is more likely.
3. Is this a reasonable assumption?
Yes. Even if [Arsenal] are more confident and [Chelsea] are trying harder, those equal out so the chance would stay at 50-50.
4. What are the probabilities of each result according to Alison's model?
In Alison's model the team who scores first is twice as likely to score the next goal.
The chance of ending 2-0, with the teachers winning, is:
$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$
so there would be a 33.3% chance of that occurring.
The chance of ending 2-0, with [Arsenal] winning, is also 33.3%:
$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$
The chance of ending 1-1, with [Arsenal] scoring the first goal and [Chelsea] scoring the second goal, is:
$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$
The chance of ending 1-1, with [Chelsea] scoring the first goal and [Arsenal] scoring the second goal, is also:
$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$
So altogether the chance of ending 1-1 is also 33.3%:
$\frac{1}{6}$+$\frac{1}{6}$=$\frac{1}{3}$
Therefore, using Alison's model there is a 33.3% chance of any of the three results occurring.
5. What are the probabilities of each result according to Charlie's model?
In Charlie's model, after a team scores, the opposing team are twice as likely to score the next goal, because they start trying harder.
The chance of ending 2-0, with [Arsenal] winning, is:
$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$
so there would be a 16.7% chance of that occurring.
The chance of ending 2-0, with [Chelsea] winning, is also 16.7%:
$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$
The chance of ending 1-1, with [Arsenal] scoring the first goal and [Chelsea] scoring the second goal, is:
$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$
The chance of ending 1-1, with [Chelsea] scoring the first goal and [Arsenal] scoring the second goal, is also:
$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$
So altogether the chance of ending 1-1 is 66.6%:
$\frac{1}{3}$+$\frac{1}{3}$=$\frac{2}{3}$
Therefore, using Charlie's model there is a much greater chance of ending up with a 1-1 result.
Krystof from Uhelny Trh, Prague, used tree diagrams to work out the probabilities.
Teachers' Resources
Why do this problem?
This problem offers the opportunity to use probability in an authentic sporting context. It introduces the idea of using a probability model to make predictions, and then to refine the model using real data.Possible approach
Introduce the idea of two equally matched teams (teams who are equally likely to score the next goal).
"If two goals are scored in a match, what different results are possible?"
"What are the probabilities of the different results?"
Give students time to work out the probabilities. While they are working, circulate and see what methods are being used.
Bring the class together and share different approaches (this may include approaches based on the incorrect assumption that win, loss and draw are equally likely).
With some classes, it may be appropriate to simulate the matches using dice. For example, the even numbers could correspond to one team scoring and the odd numbers could correspond to the other team scoring.
Next, introduce Charlie's and Alison's models, and allow some time for students to discuss their "gut feelings" about which might be more accurate, based on their own experiences.
Then allow time for students to simulate the matches using dice again, or to work out the probabilities using one of the methods discussed earlier (perhaps drawing attention to the efficiency of tree diagrams).
Students may be curious to know how accurately the models reflect reality; the possible extension below suggests how this can be explored.
Key questions
Why are the results "win", "draw" and "lose" not all equally likely?
Is the second goal independent of the first goal?
What data could be collected to evaluate the models?
Possible extension
Students could find data (from other Arsenal matches, or perhaps from school or local teams) and decide on criteria for identifying "closely matched" teams. Then matches between closely matched teams where exactly two goals were scored can be analysed to see which model best fits the data.
Students could refine the models based on the data, and could critique the models and the assumptions made in evaluating them.