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.
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
"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
Why are the results "win", "draw" and "lose" not all equally
Is the second goal independent of the first goal?
What data could be collected to evaluate the models?
Students could find data (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
Students could refine the models based on the data, and could
critique the models and the assumptions made in evaluating
Ask pairs of students to simulate the different models using dice,
and collect together all the results. Then introduce tree diagrams
to explain the results of the simulation.