We elaborate on an alternative representation of
conditional probability to the usual tree diagram. We term the representation “turtleback
diagram” for its resemblance to the pattern on turtle shells. Adopting the set
theoretic view of events and the sample space, the turtleback diagram uses
elements from Venn diagrams—set intersection, complement and partition—for
conditioning, with the additional notion that the area of a set indicates
probability whereas the ratio of areas for conditional probability. Once parts
of the diagram are drawn and properly labeled, the calculation of conditional
probability involves only simple arithmetic on the area of relevant sets. We
discuss turtleback diagrams in relation to other visual representations of
conditional probability, and detail several scenarios in which turtleback
diagrams prove useful. By the equivalence of recursive space partition and the tree,
the turtleback diagram is seen to be equally expressive as the tree diagram for
abstract concepts. We also provide empirical data on the use of turtleback diagrams
with undergraduate students in elementary statistics or probability courses.
Falk, R. (1986) Conditional Probabilities: Insights and Difficulties. Proceedings of the Second International Conference on Teaching Statistics, The Second International Committee on Teaching Statistics, Victoria, 292-297.
Tarr, J.E. and Jones, G.A. (1997) A Framework for Assessing Middle School Students? Thinking in Conditional Probability and Independence. Mathematics Education Research Journal, 9, 39-59. https://doi.org/10.1007/BF03217301
Tarr, J.E. and Lannin, J.K. (2005) How Can Teachers Build Notions of Conditional Probability and Independence? In: Jones, G.A., Ed., Exploring Probability in School, Springer, US, New York, 215-238. https://doi.org/10.1007/0-387-24530-8_10
Presmeg, N.C. (2006) Research on Visualization in Learning and Teaching Mathematics. In: Gutierrez, A. and Boero, P., Eds., Handbook of Research on the Psychology of Mathematics Education, Sense Publishers, Rotterdam, 205-235.
Sloman, S.A., Over, D., Slovak, L. and Stibel, J.M. (2003) Frequency Illusions and Other Fallacies. Organizational Behavior and Human Decision Processes, 91, 296-309. https://doi.org/10.1016/S0749-5978(03)00021-9
Dasgupta, S. and Freund, Y. (2008) Random Projection Trees and Low Dimensional Manifolds. 40th ACM Symposium on Theory of Computing, Victoria, 17-20 May 2008, 537-546. https://doi.org/10.1145/1374376.1374452
Yan, D., Huang, L. and Jordan, M. (2009) Fast Approximate Spectral Clustering. Proceedings of the 15th International Conference on Knowledge Discovery and Data Mining, Paris, 28 June-1 July 2009, 907-916.