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Probability level 8 NZC AS91585 Apply probability concepts in solving problems NZC level 8 Investigate situations that involve elements of chance • calculating probabilities of independent, combined, and conditional events AS 3.13 Apply probability concepts in solving problems WHAT ISN’T INCLUDED? AS 3.13 Apply probability concepts in solving problems WHAT ISN’T INCLUDED? No combinations, no permutations, No formal questions on expected value. This means that students are expected to bring an understanding of expected value from NCEA level 1 and 2. They may need to calculate a simple expected value as part of solving a problem, but will not be asked questions such as “calculate the expected value”. AS 3.13 Apply probability concepts in solving problems WHAT IS INCLUDED? Methods include a selection from those related to: true probability versus model estimates versus experimental estimates randomness independence mutually exclusive events conditional probabilities probability distribution tables and graphs two way tables probability trees Venn diagrams. AO S8-4 TKI A. Calculating probabilities of independent, combined, and conditional events: Students learn that some situations involving chance produce discrete numerical variables, that situations involving real data from statistical investigations can be investigated from a probabilistic perspective. These have probability distributions. They can be investigated by making assumptions about the situation and applying probability rules and/or by doing repeated trials of the situation and collecting frequencies. AO S8-4 on TKI Selects and uses appropriate methods to investigate probability situations including experiments, simulations, and theoretical probability, distinguishing between deterministic and probabilistic models. Interprets results of probability investigations, demonstrating understanding of the relationship between true probability (unknown and unique to the situation), model estimates (theoretical probability), and experimental estimates. Selects and uses appropriate tools to solve problems in probability, including two-way tables, Venn diagrams, and tree diagrams, including combined events. Solves probability problems involving conditional probabilities, randomness, independence, and mutually exclusive events. Randomness Students are expected to be familiar with the behaviour of random variables and the appearance of random distributions. What is randomness? What does randomness look like? How can we teach it? How can we assess it? Randomness What is randomness? • a lack of pattern or predictability in events What does randomness look like? file://localhost/Users/marionsteel/Desktop/workshops Is this random scatter? /probability workshop/random scatter.xls Or is this random scatter? How can we teach an understanding of randomness? Lots of hands on experience with random variables Games like Fooling the teacher Encourage students to confront their own misconceptions and fallacies about probability and randomness Teach it from year 9 onwards so that students have developed a sound understanding of it by the time they reach year 13. How can we assess “methods relating to” randomness? Methods relating to randomness are virtually all the methods of probability and statistics. Students might be asked to justify strategies or decisions, which might include reference to random outcomes or probabilities of random variables. true probability versus model estimates versus experimental estimates What is the probability that the next baby born in NZ will be a boy? We start with a basic model based on our previous knowledge and experience. With more information, we can improve our model. true probability versus model estimates versus experimental estimates What is the probability that a biased coin will land heads up? We start with a model (null hypothesis) of landing equally likely on heads and tails. We look at data, asking the question whether it provides evidence that our model is not a good representation of the real world. Deterministic and probabilistic models A deterministic model does not include elements of randomness. Every time you run the model with the same initial conditions you will get the same results. A probabilistic model does include elements of randomness. Every time you run the model, you are likely to get different results, even with the same initial conditions. Waiting times A simple model of a cash machine • Customers arrive every two minutes, on average. • Customers take 2 minutes to use the machine. • What is the probability that a customer has to wait 3 minutes or more? Waiting times In a deterministic model people arrive every two minutes and use the machine. There is no waiting time. We can use a simulation to investigate waiting times for a probabilistic model. We can simulate 15 random arrival times in a 30 minute period: 2 4 5 5 10 11 12 15 16 19 20 24 29 29 29 Modelling waiting times Modelling waiting times From our simulation, 2/15 customers waited 3 minutes or more. Our estimate of the probability that a customer waits 3 minutes or more is 0.13. The following slides are from Auckland Statistics Day 2004 (apologies to the University of Auckland Statistics Department as their logo wouldn’t copy). Since 2004, we have been encouraged to deemphasize Venn diagrams for solving probability problems. Two way tables are a much more effective problem solving method, and should be student’s first choice. What progress has been made towards that shift in teaching practice? House Sales What proportion of the houses that sold for over $600,000 were on the market for less than 30 days? Days on the market Selling price Less than 30 days 30 - 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 8 4 0 12 82 80 19 181 Over $600,000 Total House Sales What proportion of the houses that sold for over $600,000 were on the market for less than 30 days? Days on the market Selling price Less than 30 days 30 - 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 8 4 0 12 82 80 19 181 Over $600,000 Total House Sales What proportion of the houses that sold for over $600,000 were on the market for less than 30 days? Days on the market Selling price Less than 30 days 30 - 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 8 4 0 12 82 80 19 181 Over $600,000 Total House Sales What is the probability a house sold for under $300,000 given that it sold in less than 30 days? Days on the market Selling price Less than 30 days 30 - 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 8 4 0 12 82 80 19 181 Over $600,000 Total House Sales What is the probability a house sold for under $300,000 given that it sold in less than 30 days? Days on the market Selling price Less than 30 days 30 - 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 8 4 0 12 82 80 19 181 Over $600,000 Total House Sales What is the probability a house sold for under $300,000 given that it sold in less than 30 days? Days on the market Selling price Less than 30 days 30 - 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 8 4 0 12 82 80 19 181 Over $600,000 Total Solving probability problems Encourage the use of a two way table as the first method to consider. Encourage flexibility. Solve the same problem using: Two way tables Tree Venn diagram Probability algebra