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Visualizing Chance: Tackling Conditional Probability Misconceptions

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Topics and Trends in Current Statistics Education Research

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

Probabilistic reasoning is essential for operating sensibly and optimally in the 21st century. However, research suggests that students have many difficulties in understanding conditional probabilities and that Bayesian-type problems are replete with misconceptions such as the base rate fallacy and confusion of the inverse. Using a dynamic pachinkogram, a visual representation of the traditional probability tree , we explore six undergraduate probability students’ reasoning processes as they interact with this tool. Initial findings suggest that in simulating a screening situation, the ability to vary the branch widths of the pachinkogram may have the potential to convey the impact of the base rate. Furthermore, we conjecture that the representation afforded by the pachinkogram may help to clarify the distinction between probabilities with inverted conditions.

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Notes

  1. 1.

    The word pachinkogram originates from the Japanese pachinko machine which resembles a vertical pinball machine.

  2. 2.

    Note +ve denotes a positive test result. Therefore \(P({\text{Diabetic}}\,|\,{{ {+}\text{ve})}}\) is the probability of having diabetes, given that a person has a positive test result and \(P({{{+}\text{ve}}}\,|\,{\text{Diabetic}})\) is the probability of having a positive test result, given that a person is diabetic.

  3. 3.

    Students were provided with a table of data from Coppell et al. (2013) from which they could determine the base rate for the subgroup of the population referred to in the questions.

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Acknowledgements

This work is supported by a grant from the Teaching and Learning Research Initiative (http://www.tlri.org.nz/).

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Authors and Affiliations

  1. Department of Statistics, The University of Auckland, Auckland, New Zealand

    Stephanie Budgett & Maxine Pfannkuch

Authors
  1. Stephanie Budgett
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  2. Maxine Pfannkuch
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Correspondence to Stephanie Budgett .

Editor information

Editors and Affiliations

  1. Program in Mathematics Education, Michigan State University, East Lansing, MI, USA

    Gail Burrill

  2. Faculty of Education, University of Haifa, Haifa, Israel

    Dani Ben-Zvi

Appendix: Some of the Diabetes Task Questions

Appendix: Some of the Diabetes Task Questions

A new housing development has been built in your neighbourhood. In order to service the needs of this new community, a new health clinic has opened. As part of the health clinic’s enrolment procedure, new patients are required to undergo health check-ups which include, among other things, a series of blood tests. One such test is designed to measure the amount of glucose in an individual’s blood. This measurement is recorded after the individual fasts (abstains from eating) for a prescribed period of time. Fasting blood glucose levels in excess of 6.5 mmol/L are deemed to be indicative of diabetes. This threshold of 6.5 mmol/L works most of the time with about 94% of people who have diabetes being correctly classified as diabetics and about 98% of those not having diabetes being correctly classified as non-diabetics.

The prevalence of diabetes in the NZ population is about 7% (i.e. approximately 7% of the NZ population are estimated to have diabetes).

figure a

Graph above adapted from Pfannkuch et al. (2002, p. 28)

  • Question 1

    1. (a)

      As part of enrolment in this health clinic, an individual has a fasting blood test. He/she is told that his/her blood glucose level is higher than 6.5 mmol/L. What are the chances that he/she has diabetes? Provide an intuitive answer.

    2. (b)

      Now use the software tool to answer Question 1 (a).

    3. (c)

      Reflecting on your answer to (b), how does this compare with your answer to (a)?

  • Question 2

    1. (a)

      As part of his enrolment in this health clinic, a male aged between 65 and 74 has a fasting blood test.Footnote 3 He is told that his blood glucose level is higher than 6.5 mmol/L. What are the chances that he has diabetes? Provide an intuitive answer.

    2. (b)

      Now use the software tool to answer Question 2 (a).

    3. (c)

      Reflecting on your answer to (b), how does this compare with your answer to (a)?

  • Question 3

    1. (a)

      As part of their enrolment in this health clinic, a person of Pacific ethnicity and aged over 75 has a fasting blood test. He/she is told that his/her blood glucose level is higher than 6.5 mmol/L. What are the chances that he/she has diabetes? Provide an intuitive answer.

    2. (b)

      Now use the software tool to answer Question 3 (a).

    3. (c)

      Reflecting on your answer to (b), how does this compare with your answer to (a)?

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Budgett, S., Pfannkuch, M. (2019). Visualizing Chance: Tackling Conditional Probability Misconceptions. In: Burrill, G., Ben-Zvi, D. (eds) Topics and Trends in Current Statistics Education Research. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-030-03472-6_1

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