Which team will win?

9546 /www/nrich/html/content/id/9546/ 1 2012-10-10T12:06:28 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="framework"><mdo:image alt="animal playing football" src="AnimalPlayingFootball.png" style="width: 250px; height: 250px; margin: 10px; float: right;"></mdo:image>Every weekend, Team Beaver and Team Yeti play each other at 2-Goal Football - they play until two goals have been scored. Sometimes games last only a minute or two, sometimes they seem to go on for ever. Over a 36-week season, which team is likely to win more games? Team Beaver or Team Yeti?</div> You are going to investigate this question through a practical experiment. If possible use a die with four yellow and two blue faces (if this is not available, use an ordinary die and let 1, 2, 3 and 4 correspond to a yellow face, and 5 and 6 to a blue face.) First investigate what happens for one game: <ol> <li>Throw the die. A yellow (1, 2, 3, 4) face means a goal to Team Yeti, a blue (5, 6) face means a goal to Team Beaver. Record this on the Results Worksheet - you might find it helpful to use a yellow colour for Team Yeti and a blue colour for Team Beaver.</li> <li>Throw the die again, to see who scores the second goal, and record it on the <a href="/content/id/9546/WhichTeamWillWin_worksheet1.pdf">Results Worksheet</a>.</li> </ol> Who won? Compare your result with other people's - any surprises? Now repeat the experiment 36 times in total, to get the results for a full season. Are you surprised by your results? How do they compare with what you would expect? Which team scores most often? Why? Does this mean they will always win? Why not? How often do you expect the Yetis to score? How often do you expect the Beavers to score? Did the Yetis win the proportion of games you would expect? How about the Beavers? If not, can you explain what's happening? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed">Which team will win? Every weekend, Team Beaver and Team Yeti play each other at 2-Goal Football - they play until two goals have been scored. Sometimes games last only a minute or two, sometimes they seem to go on for ever. Over a 36-week season, which team is likely to win more games? Team Beaver or Team Yeti?</div> <h3> Why do this problem?</h3> <a href="http://nrich.maths.org/9546">This problem</a><a href="http://nrich.maths.org/9546"> </a>provides an introduction to probability, through experimentation and discussion. The football game is limited to exactly two goals, so that modelling it is straight-forward. <h3>Possible approach</h3> 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. 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! <h3>Key questions</h3> Who is likely to score more often? Why? Does this mean they will always win? Why not? Are any of the results in the tally interesting or surprising? Why? Why isn't YB the same as BY? (This could focus on the psychological difference between scoring first, or coming back from a losing position). <h3>Possible approach (continued)</h3> 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. <a href="/content/id/9546/WhichTeamWillWin_worksheet1.pdf">This worksheet</a> 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. Students should be encouraged to consider how their results over 36 games compare with the initial class tally, and any initial conjectures could be re-considered at this stage - but encourage students not to think that there are any definitive answers as yet. When students have a full set of results, they should transfer them to the tree diagram and 2-way table on <a href="/content/id/9546/WhichTeamWillWin_Worksheet2.pdf">this worksheet</a>. 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. <h3>Key questions</h3> 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.) <h3>Possible extension</h3> 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? <h3>Possible support</h3> 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.</mdoxml> 5 3 0 0 1 0 0 Which team will win? A practical experiment which will introduce students to tree diagrams, and help them to understand that outcomes may not be equally likely. jag55 work in progress jag55 Probability - modelling approach jag55 ready for simulation