Using frequencies in tree diagrams to calculate probabilities


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1 Lesson Element Using frequencies in tree diagrams to calculate probabilities Instructions and answers for teachers These instructions should accompany the OCR resource 'Using frequencies in tree diagrams to calculate probabilities activity which supports OCR Level 3 Certificate in Quantitative Problem Solving (MEI) and Level 3 Certificate in Quantitative Reasoning (MEI). This activity offers an opportunity for maths skills development. Associated materials: 'Using frequencies in tree diagrams to calculate probabilities Lesson Element learner activity sheet and 'Using frequencies in tree diagrams to calculate probabilities supporting PowerPoint.
2 This resource was inspired by a page from David Speigelhalter s website at which is a great open resource with links and discussions. An accompanying PowerPoint file is provided for your use and these instructions contain references to the relevant slides. Introduction In the OCR/MEI Quantitative Reasoning specification, it states that candidates should be able to work with a tree diagram when calculating or estimating a probability, including conditional probability. Learners can choose whether to work with either frequencies or probabilities in tree diagrams.. This lesson element also considers using frequencies in tables and Venn diagrams as these can be a good starting point for understanding their use in tree diagrams. (Slides 3, 4 and 5) The idea of using frequencies is very simple: instead of saying the probability of X is 0.20 (or 20%), we would say out of 100 situations like this, we would expect X to occur 20 times. This simple reexpression can prevent confusion and make probability calculations easier and more intuitive for learners by clarifying what a probability means. When we hear the phrase the probability it will rain tomorrow is 30%, what do we mean? That it will rain 30% of the time? Over 30% of the area? In fact it means that out of 100 such computer forecasts, we can expect it to rain after 30 of them. By clearly stating what the denominator is ambiguity is avoided. It has been shown that, by using frequencies, people find it easier to carry out nonintuitive conditional probability calculations. This is the standard approach taught to medical students for risk communication, and is used extensively in public information, for example, in breast screening leaflets. (Slide 6) These guidance notes should be used with the PowerPoint resource (Using frequencies in tree diagrams to calculate probabilities) and the student worksheet which provides additional questions for learners to try individually or in pairs/groups. The intention is that students will discuss the questions in pairs or groups and present their methods to the rest of the group. They could also be asked to create similar but different problems to set the group. The slides can be used as back up to demonstrate approaches that the students do not come up with themselves for example, students may initially not be aware that using frequencies is a good method or that they could use a Venn diagram or table.
3 Task 1 Basic probability This is essentially a one level tree. Questions can involve going either from probabilities (expressed as decimals, fractions and % s) to expected frequencies, or vice versa. The problems can be drawn as either expected frequency or probability trees. Do not provide any guidance at first use the questions to assess prior learning and preferred methods and explanations. 1 Some balanced dice have probability of coming up 4. Out of 60 throws, how many 4 s would 6 we expect to come up? (Slides 7 and 8 show two approaches) 80% of the school students can roll their tongues. If I pick 1000 students at random, how many do you expect will NOT be able to roll their tongues? (Slides 9 and 10 show two approaches) In a typical school with 80 Year 10 students, 64 of them will have a profile on the social media site Faceache. What is the probability that if we pick a Year 10 student at random, they will not have a profile? (Slide 11 and 12 show a possible approach) Task 2 Comparison of probabilities This involves comparison of two different situations, and can be represented using a pair of trees. It is ideal for dealing with challenging and realistic questions concerning relative and absolute risks. Discussion questions students work in pairs or groups and present alternative solutions: A newspaper headline says that eating radishes doubles your chance of getting Smith s Disease. 1% of people who don t eat radishes get Smith s Disease anyway. o Out of 200 people not eating radishes, how many would I expect to get Smith s disease? o Out of 200 people eating radishes, how many would I expect to get Smith s disease? o How many people have to eat radishes, in order to get one extra case of Smith s disease? (Slides 13 and 14 show two possible approaches)
4 Task 3 Conditional probabilities This requires twolevel trees, and can also bring in twoway tables and Venn diagrams. First, given the conditional probabilities, set up the frequency tree. Then calculate the expected frequencies and convert back to probabilities if wanted. Discussion questions students work in pairs or groups and present alternative solutions: Ask students to write a question based on the probability tree diagram below: Possible question: A weather forecast is generally right. When it forecasts rain, 90% of the time it rains. When it forecasts no rain, 70% of the time it does not rain. In a typical September they forecast rain on twothirds of days and no rain on onethird of days. Draw a frequency tree diagram based on the information given. How many days would you expect it to rain each September? What is the probability that a random day in September is not rainy? From the expected frequency tree, we expect it to rain on a total of = 21 days in September, and 9 not rain in 9. So the probability that a random day in September is not rainy is =
5 Task 4 Reverse conditional probabilities This is where things can get a bit tricky, but using expected frequency representations allows students to tackle some of the classic nonintuitive probability problems essentially Bayes theorem. If they can do these, they have learnt a subtle and valuable skill. Weather forecasting: of the times it rains, what proportion did the forecast get it right? It rains 21 times, and in 18 the rain was forecast, so the proportion is 18 6 = : ie when it rains, 21 7 there is 7 6 chance that the rain was forecast. Using frequencies is essentially the same as, but easier to understand than, finding the same chance using probabilities, and much easier than interpreting the formula P(A B) = P(A B)! P(B) We d like to know your view on the resources we produce. By clicking on the Like or Dislike button you can help us to ensure that our resources work for you. When the template pops up please add additional comments if you wish and then just click Send. Thank you. OCR Resources: the small print OCR s resources are provided to support the teaching of OCR specifications, but in no way constitute an endorsed teaching method that is required by the Board, and the decision to use them lies with the individual teacher. Whilst every effort is made to ensure the accuracy of the content, OCR cannot be held responsible for any errors or omissions within these resources. We update our resources on a regular basis, so please check the OCR website to ensure you have the most up to date version. OCR This resource may be freely copied and distributed, as long as the OCR logo and this message remain intact and OCR is acknowledged as the originator of this work. OCR acknowledges the use of the following content: Maths icon: Air0ne/Shutterstock.com, Thumbs Up and down icons: alexwhite/shutterstock.com
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