This problem models the interpretation of statistics for testing for eg. cancer, HIV, pregnancy, DNA at a crime scene, and many other similar situations.
Rather than focusing on these potentially difficult topics, however, we have chosen a 'jokey' theme so that students can get into the maths, then appreciate the real world applications through discussion of what the results mean.
Students find tree diagrams, and their interpretation, difficult. The approach used in this problem will help them to become more familiar with tree diagrams, and interpreting them.
Put students into groups of 3 or 4, and have each group collect a 6-sided die, and some red, blue, green and yellow multi-link cubes. We would suggest you don't hand out any worksheets until students have collected their data (pairs of cubes) and had initial discussion of the results both in their groups and in the class as a whole.
Take the students through the scenario for one student. Then get them to collect data - pairs of multi-link cubes - for 36 students. You could use this presentation to start the activity off:
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Following discussion of the results, students could then record their results on the tree diagram and in the 2-way table provided in this worksheet.
How do the experimental results compare with what we would expect?
How many students in total are accused? Are you surprised? Why (not)?
How many innocent students are accused? What proportion of innocent students are accused? So what is the probability that an truthful student is accused? Are you surprised? Why (not)?
Is the probability that a truthful student is accused the same as the probability that an accused student is telling the truth?
What figures are in both the tree diagram and the 2-way table, which are not?
What are the advantages of the tree diagram? How about the 2-way table? What information can you find easily from each, what is more difficult or impossible to find?
What difference would it make if Mr D wasn't always right about students who are lying. How would you change the model?
Critique the model. What assumptions does it make? How could you improve it, to make it more realistic?
All students should be able to carry out the experiment, once they have understood the scenario and done an initial trial all together. The worksheet is designed to help students to interpret their results. Students who find the questions above difficult could instead focus on the comparison between experimental and expected results, and on critiquing the model - for instance, is it reasonable to suppose that 1/6 of students would lie about their homework? If not, how could we change the experiment?