### Abstract:

The purpose of this descriptive case study analysis was to provide portraits of the heuristics students used and difficulties they encountered solving conditional probability problems prior to and after two-week instruction on sample space, probability, and conditional probability. Further analysis consisted of evaluating the data in relation to a previously designed Conditional Probability Framework for assessing students levels of thinking developed by Tarr and Jones (1997). Five volunteer participants from a contemporary college mathematics course participated in pre-and post-interviews of a Probability Knowledge Inventory. The Inventory consisted of seven tasks on sample space, probability, and conditional probability. The semi-structured interviews provided participants' explanations on the development of their solutions to the seven tasks.
Among the five participants, rationalizing, finding the odds, computing the percentages, and stating the ratio of a problem were the preferred heuristics used to solve the problems on the Probability Knowledge Inventory. After the two-week instruction, two of the four participants who did not previously use computation of probability to solve the problem changed their use of heuristics. The difficulties the students encountered prior to instruction included understanding the problem; recognizing the original sample space and when it changes; lacking probability vocabulary knowledge; comparing probability after the sample space changed; understanding the difference between probability and odds; and interchanging ratio, odds, and percentages-sometimes incorrectly-to justify their solution. After the two-week instruction, the students' difficulties diminished. Some improvements included a greater ability to understand the question of interest, to recognize the change in the sample space after a conditioning event, to use probability terminology consistently, and to compare probability after the sample space has changed.
Comparisons to the Probability Framework revealed that four of the five participants exemplified Level 3 thinking-being aware of the role that quantities play in forming conditional probability judgements. One participant exemplified a Level 4 thinking-being aware of the composition of the sample space, recognizing its importance in determining conditional probability and assigning numerical probabilities spontaneously and with explanation.