# When 95% Accurate Isn t Written by: Todd CadwalladerOlsker California State University, Fullerton

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1 When 95% Accurate Isn t Written by: Todd CadwalladerOlsker California State University, Fullerton Overview of Lesson In this activity, students will investigate Bayes theorem using simulated data (generated by a calculator), expected frequencies, and probabilities. By progressing through these perspectives in a deliberate order (first simulated data, then expected frequencies, ending with probabilities) students will discover Bayes theorem for themselves. Along the way, they will be surprised by the role that false positives play in determining conditional probabilities. GAISE Components This investigation somewhat follows the four components of statistical problem solving put forth in the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report. The four components are: formulate a question, design and implement a plan to collect data, analyze the data by measures and graphs, and interpret the results in the context of the original question. This is a GAISE Level B activity. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Common Core State Standards Grade Level Content (High School) S-CP. 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. S-CP. 6. Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. NCTM Principles and Standards for School Mathematics Data Analysis and Probability Standards for Grades 9-12 Understand and apply basic concepts of probability: understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases; use simulations to construct empirical probability distributions; compute and interpret the expected value of random variables in simple cases; understand the concepts of conditional probability and independent events. Prerequisites Students will need sufficient knowledge of programmable calculators to run a program to simulate data. Students may need to be able to transfer programs from one calculator to another; alternatively, the instructor can transfer the program to calculators ahead of time. Students will 1

2 need a basic understanding of expected values and conditional probabilities. Students should not have previously seen Bayes theorem, as they will discover Bayes theorem through this activity. Learning Targets Students will be able to use Bayes theorem to compute conditional probabilities based on given information. Students will be able to re-create Bayes theorem using expected frequencies. Time Required 1 class period, possibly 2 if the instructor and students want to investigate further questions. Materials Required A copy of the Activity Sheet (pages 11-14); a programmable calculator (TI-83, TI-84, or similar); the program ACPPROG for TI-84 calculators. ACPPROG is available from the STEW website as a compressed.zip file; instructions for transferring the program to TI-84 calculators are included in the.zip file. The ACPPROG code is included at the end of this lesson plan; instructors can adapt it for use in other calculators or programs, such as Excel. Instructional Lesson Plan The GAISE Statistical Problem-Solving Procedure I. Formulate Questions(s) Begin the lesson with a class discussion of the question, What does it mean to say that a test, or an observation, is 95% accurate? The instructor may wish to use specific contextual questions, such as, What does it mean to say that a weather forecaster is 95% accurate? or What does it mean to say that a spam filter is 95% accurate? This can (and should) lead to some discussion of the meaning of conditional probability statements: for example, the class could discuss the difference between a spam filter that filters 95% of the actual spam and one for which 95% of the filtered messages are spam, and how those two statements could be represented as conditional probabilities. In this example, if S is the event that an is spam and F is the event that the is filtered, a spam filter that filters 95% of the spam is one for which. On the other hand, a filter for which 95% of the filtered messages are spam is one for which. Next, the instructor should distribute the activity sheet (pages 11-14) and (if necessary) programmable calculators with the ACPPROG program loaded. The class should discuss the following oil exploration problem given at the beginning of the activity sheet: When exploring an area for oil, oil company surveyors will look for leads, formations on the Earth's surface that indicate the possibility of oil beneath the surface. However, just because it is possible to find oil beneath these leads does not mean that there is oil. Suppose that only 6% of leads actually have oil present. The oil company's surveyors will run a series of tests to determine the likelihood of the presence of oil. The surveyors' tests are 95% accurate for both the presence and the absence of oil: If oil is present, the surveyors' tests will reveal that oil is present with a probability of 0.95, and if oil is absent, the surveyors' tests will reach a negative 2

10 Possible Extensions Students can come up with their own questions that involve Bayes theorem. Any probability that can be influenced by additional information can be turned into a problem involving Bayes theorem. For example, suppose the students joke that their science and math teachers are conspiring to give quizzes on the same day. This can be turned into a Bayes theorem problem: If the science class comes before the math class, they can estimate the percentage probability that they have a math quiz given that they had a science quiz earlier in the day. The question can then be asked, Suppose you know you have a math quiz tomorrow. What is the probability that you will also have a science quiz tomorrow? Students will need to estimate the relevant numbers to solve this problem. References 1. Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report, ASA, Franklin et al., ASA, Common Core State Standards for Mathematics, Common Core State Standards Initiative (CCSSI). Released June 2, Principles and Standards for School Mathematics. National Council of Teachers of Mathematics (NCTM). Reston, VA: NCTM, This activity is based on the following: When 95% Accurate Isn t: Exploring Bayes Theorem. CadwalladerOlsker, T, Mathematics Teacher, 104(6), p ,

14 18. Summarize your answers to 16 and 17 in the table below: Positive Test Negative Test Oil Present No Oil Present (Note: The sum of all four boxes should be equal to 100% of the total leads.) 19. What percentage of leads should we expect to test positive for oil? 20. Of those leads that we expect to test positive for oil, what percentage should we expect to actually have oil present? 21. Based on your answers to 19 and 20, what is the probability that a lead has oil present given that it tests positive? In other words, what is based on these percentages? 14

15 ACPPROG Program Source Code Below is the source code for the program ACPPROG. The code below may be modified or rerewritten for use in other calculators or programs, such as Microsoft Excel. The following should be entered as five separate programs into the calculator, but only ACPPROG will be directly used. The ACPPROG.zip file also includes the programs ACPPROGS, DISPDATC, and DISPPRB. ACPPROGS is a solutions version of the program that displays the conditional probabilities and based on the generated data, as well as the theoretical probabilities calculated by Bayes Theorem. PROGRAM:ACPPROG :prgmgetprb :prgmcalcprb :prgmgendata :prgmdispdata :Pause :ClrHome PROGRAM:GETPRB :ClrHome :Input "P(A)? ",A :Input "P(B,A)? ",B :Input "P(-B,-A)? ",C PROGRAM:CALCPRB :(A*B) K :(A*(1-B)) L :((1-A)*(1-C)) M :((1-A)*C) N :(K/(K+M)) Q :(N/(N+L)) R PROGRAM:GENDATA :ClrHome :Input "NO. OF DATA? ",I :0 S :0 T :0 U :0 V :0 X :For(X,1,I) :rand Y :If Y<K:Then :S+1 S:Else :If Y<(K+L):Then :T+1 T:Else :If Y<(K+L+M):Then :U+1 U:Else 15

16 :V+1 V :End:End:End:End PROGRAM:DISPDATA :ClrHome :Output(1,1,"GENERATED DATA:") :Output(2,1,"A+B:") :Output(2,10,S) :Output(3,1,"A+-B:") :Output(3,10,T) :Output(4,1,"-A+B:") :Output(4,10,U) :Output(5,1,"-A+-B") :Output(5,10,V) 16

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