Being representative

9843 /www/nrich/html/content/id/9843/ 1 2013-01-31T18:06:31 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="framework"><mdo:image alt="debating" src="debating-1.png" style="width: 250px; height: 215px; margin: 10px; float: right;"></mdo:image> Ms Dickens decides that her English class should practise their debating skills. The motion is to be: "This class believes it is a good idea for students to learn poems by heart." But when she asks who wants to speak in the debate, she has far too many volunteers, so she suggests that each side should choose one person to speak for them. However, Ingrid, says this is unfair. There are more people on her side than the other, so her chance of being chosen is less than theirs. There are 4 people who want to speak on behalf of the motion and 12 who want to speak against it, but only time for 2 speakers. How are the speakers to be chosen?</div> <h3>Ingrid's first suggestion</h3> "Give everyone who wants to speak a number. Then use the random number generator on a calculator to choose two people." Start by modelling this method. Put 12 counters (or cubes) in one colour and 4 in another in a non-transparent bag. Draw out one counter and note which colour it is. Then put it back and draw out another, noting which colour it is. How does this model Ingrid's suggestion? Why does the counter need to be put back in the bag? What was the result in your trial? If you do several trials, do the results appear to model the balance of opinion in the class in a fair way? Does the choice of the second speaker depend on the first at all? What is the chance of both sides being represented? You should use a tree diagram to help you investigate this. What are the problems with this model? <h3>Ingrid's second suggestion</h3> "Put the name of everyone who wants to speak in a hat. Then draw out two names." Model this method. Put 12 counters (or cubes) in one colour and 4 in another in a non-transparent bag. Draw out one counter and note which colour it is - do not return it to the bag this time. Draw out a second counter, noting which colour it is. How does this model Ingrid's second suggestion? Why is the counter not put back in the bag this time? What was the result in your trial? Does the choice of the second speaker depend on the first at all? If you do several trials, do the results appear to model the balance of opinion in the class in a fair way? What is the chance of both sides being represented? You should use a tree diagram to help you investigate this. How does your tree diagram for this method differ from your first one? What are the problems with this model? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed">Being Representative Ms Dickens decides that her English class should practise their debating skills. The motion is to be: "This class believes it is a good idea for students to learn poems by heart." But when she asks who wants to speak in the debate, she has far too many volunteers, so she suggests that each side should choose one person to speak for them. However, Ingrid, says this is unfair. There are more people on her side than the other, so her chance of being chosen is less than theirs. There are 4 people who want to speak on behalf of the motion and 12 who want to speak against it, but only time for 2 speakers. How are the speakers to be chosen?</div> <h3> </h3> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/9843">This problem</a> provides a context for exam questions which start 'A bag contains 12 white balls and 4 black ones ...' By focusing purely on the technicalities of probability calculation,s questions like this obscure the fact that sampling methods are very important. How a sample is chosen can skew results quite dramatically. This problem is designed to provide a scenario where sampling with and without replacement can be investigated, and the merits of each discussed alongside the merits of choosing one representative for each point of view. <h3>Possible approach</h3> Put students into groups of 3 or 4, and have each group collect the equipment they need. Take the students through the scenario for Ingrid's first suggestion - sampling with replacement. Discuss how the model relates to what Ingrid has suggested. Then get everyone to try it 16 times, putting their results onto a tree diagram. Results from the class could also be collated to see what happens when there is a lot of data. Then model Ingrid's second suggestion. Discussion could focus around the advantages and disadvantages of each method, compared with simply choosing one person to represent each point of view. Discussion should then lead into analysing the two methods to see if what appears to be the case can be substantiated. Students should draw tree diagrams for each method, with the first set of branches representing the first counter drawn, and the second set representing the second counter drawn. <a href="/content/id/9843/BeingRepresentative_worksheet.pdf">This worksheet</a> could be used to help with this. <h3>Key questions</h3> What is different about sampling with and without replacement? This is an opportunity to make explicit the difference between independent and dependent events, and that the probability of the second event changes if it is dependent on the first, but not if it is independent. For each suggested method, do the number of potential speakers change at all after the first speaker is chosen? Why is it good practice to sum calculated probabilities for all the outcomes as a check? Why must the sum of the probabilities on any pair of branches on a probability tree always sum to 1? <h3>Possible extension</h3> How does it change things if the numbers of potential speakers changes? Why are the probabilities the same for FA and for AF, regardless of whether there is replacement or not? Critique the models. 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 one or more initial trials all together. The <a href="/content/id/9843/BeingRepresentative_worksheet.pdf">worksheet</a> is designed to structure the analysis to help those who find it difficult to calculate the probabilities from the experiment.</mdoxml> 5 3 0 0 0 1 0 Being representative How can you make sure that two sides of an argument are fairly represented when choosing speakers for a debate? jag55 Probability - modelling approach