A practical experiment which uses tree diagrams to help students understand the nature of questions in conditional probability.
This problem provides an introduction to probability, through experimentation and discussion. It covers the concepts appropriate for students' first formal lessons on probability. The football game is limited to exactly two goals, so that modelling it is straight-forward.
Group the students into threes as far as possible. Give each group one probability die if available (a die with four yellow sides and two blue ones - these can be made by sticking coloured dots onto plain dice). Otherwise, groups should be provided with one normal die - in this case 1, 2, 3 and 4 correspond to a yellow face, and 5, 6 to a blue face.
Explain the scenario, and ask each group to throw their die, to simulate one football game.
The game is deliberately set up so that Team Yeti are more likely to score than Team Beaver. Rather than drawing attention to this, give students a chance to observe that the game "isn't fair", and use this to discuss that perhaps Team Yeti are a much stronger team...
Tally the results around the class under the headings of YY (2-0 to the Yetis), BB (0-2 to the Beavers), YB (1-1 draw, with the Yetis scoring first), BY (1-1 draw with the Beavers scoring first) - it is important not to amalgamate the draws, as the two cases are not the same!
Who is likely to score more often? Why?
Does this mean they will always win?
Are any of the results in the tally interesting or surprising? Why?
What are the different ways of getting a draw?
It is best to avoid asking students what they expect to happen at this stage so that guessing is not encouraged, but if they do have ideas about what they might expect, they should be written up on the board, to be considered further when more data is available.
The initial discussion should be followed by each group getting a set of 36 results, which models a 36 week season. This worksheet could be used for them to record their results in a tally table. Using coloured pens (yellow and blue) will help them to get a feel for the pattern as it emerges, and will also help them to
collate their results.
When students have a full set of results, they should transfer them to the tree diagram and 2-way table on this worksheet. They should then fill in the second tree diagram and 2-way table for what they would expect to happen. There are also questions which will help them to examine what happened
compared to what we would expect to happen.
This problem can also be explored using the interactive environment.
Revisit the tally and any initial conjectures. Does the evidence support them or suggest that they should be discarded?
What proportion of games did the Yetis (Beavers) win? What proportion would we expect them to win?
Is the proportion of games won by the Yetis (Beavers) the same as the probability that they will score a goal? Why not?
Considering just the draws, who scored first most often? (Intuition may well suggest that because the Yetis are more likely to score, there will be more YB draws than BY draws, but the evidence may well not support this.)
What do you think would happen if the Yetis were three times as likely to score as the Beavers? How would this change the expected results?
All students should be able to collect the data for 36 games. Those who have difficulty in transferring the data from their tally to the tree diagram and 2-way table could be helped by doing it as a whole class discussion, using large diagrams on the board, and the data either for one group, or for the whole class.