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Lessons
Time-Axis Fallacy and Bayes' Theorem
Discipline
Mathematics
Grades
10, 11, 12
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Professional Commentary

Most students understand that the probability of an event occurring can be influenced by another event that has already occurred. However, many students do not understand that the probability of an event occurring can actually be dependent on an event that occurred later. Having information about the outcome of a later event can be used to revise probabilities of the occurrence of a previous event. This lesson on the time-axis fallacy will help students understand this important, but counterintuitive, idea by developing an understanding of Bayes' Theorem. The lesson plan includes a brief discussion of the mathematical topics, a list of needed materials, a suggested teaching procedure that includes several interesting problem scenarios, and a complete solution guide. (author/pk)


Common Core State Standards for Mathematics
High School - Statistics and Probability
Conditional Probability and the Rules of Probability
Understand independence and conditional probability and use them to interpret data
HSS-CP.A.3 
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
HSS-CP.A.5 
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model
HSS-CP.B.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.
HSS-CP.B.8 
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
Ohio Mathematics Academic Content Standards (2001)
Data Analysis and Probability Standard
Benchmarks (8–10)
J.
Compute probabilities of compound events, independent events, and simple dependent events.
Grade Level Indicators (Grade 9)
9.
Identify situations involving independent and dependent events, and explain differences between and common misconceptions about probabilities associated with those events.
10.
Use theoretical and experimental probability, including simulations or random numbers, to estimate probabilities and to solve problems dealing with uncertainty; e.g., compound events, independent events, simple dependent events.
Grade Level Indicators (Grade 11)
11.
Examine statements and decisions involving risk; e.g., insurance rates and medical decisions.
Grade Level Indicators (Grade 12)
6.
Use theoretical or experimental probability, including simulations, to determine probabilities in real-world problem situations involving uncertainty, such as mutually exclusive events, complementary events and conditional probability.
Mathematical Processes Standard
Benchmarks (11–12)
F.
Present complete and convincing arguments and justifications, using inductive and deductive reasoning, adapted to be effective for various audiences.
G.
Understand the difference between a statement that is verified by mathematical proof, such as a theorem, and one that is verified empirically using examples or data.
Principles and Standards for School Mathematics
Data Analysis and Probability Standard
Understand and apply basic concepts of probability
Expectations (9–12)
understand the concepts of conditional probability and independent events;
Reasoning and Proof Standard
Develop and evaluate mathematical arguments and proofs