The dog ate my homework!

9525 /www/nrich/html/content/id/9525/ 1 2013-01-29T11:59:03 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="framework"><mdo:image alt="the dog ate my homework" src="dog_ate_homework.png" style="width: 273px; height: 200px; margin-left: 10px; margin-right: 10px; float: right;"></mdo:image>A certain teacher, Mr L I Detector, claims he can tell when students are lying about their homework. This is true. Unfortunately, he also accuses some students who are telling the truth. So what are the chances that someone will be wrongly accused? </div> You are going to investigate this question through a practical experiment. Start by collecting a 6-sided die, and some red, blue, green and yellow multi-link cubes. First investigate what happens for one student: <ol> <li>Throw the die. <ul style=""> <li>If you get a 6, this means the student is lying about their homework. Take a red cube.</li> <li>If you get any other result, the student is telling the truth. Take a blue cube.</li> </ul> </li> <li>If you got a 6, you don't need to throw the die again (why not?). Take a yellow cube - this indicates that Mr D accuses the student of lying about their homework.</li> <li>If you didn't get a 6, throw the die again. <ul style=""> <li>A 1 means that Mr D accuses the student, even though they were telling the truth. Take a yellow cube.</li> <li>Anything else means that Mr D believes the student. Take a green cube.</li> </ul> </li> </ol> Put your two cubes together. What do each of these mean? <ul> <li>red and yellow</li> <li>blue and yellow</li> <li>blue and green</li> </ul> Why can't you have red and green? Now repeat the experiment 36 times in total. You should end up with 36 pairs of cubes. Are you surprised by your results? How do they compare with what you would expect? (You may find <a href="/content/id/9525/DogAteHomework_studentworksheet.pdf">this worksheet</a> helpful here). What proportion of truthful students are accused? So what is the probability of a truthful student being accused? Are you surprised by this? Is the probability that a truthful student is accused the same as the probability that an accused student is actually telling the truth? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed">The dog ate my homework! A certain teacher, Mr L I Detector, claims he can tell when students are lying about their homework. This is true. Unfortunately, he also accuses some students who are telling the truth. So what are the chances that someone will be wrongly accused?</div> <h3> Why do this problem?</h3> <a href="http://nrich.maths.org/9525">This problem</a> 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. <h3>Possible approach</h3> 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: <a href="http://nrich.maths.org/content/id/9525/9525_presentation.html"> Full Screen Version</a> <iframe height="540" src="http://nrich.maths.org/content/id/9525/9525_presentation.html" width="720"></iframe> <a href="/content/id/9525/DogAteMyHomework.ppt">Downloadable powerpoint version</a> Following discussion of the results, students could then record their results on the tree diagram and in the 2-way table provided in <a href="/content/id/9525/DogAteHomework_studentworksheet.pdf">this worksheet.</a> <h3>Key questions</h3> 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? <h3>Possible extension</h3> 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? <h3>Possible support</h3> 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? </mdoxml> 5 3 0 0 1 1 0 The dog ate my homework! A practical experiment which uses tree diagrams to help students understand the nature of questions in conditional probability. jag55 ready for simulation jag55 Probability - modelling approach jag55 Probability - modelling approach curriculum resources