The Birthday Problem is "How many people must be in a room before the probability that some share a birthday (ignoring the year and ignoring leap days) becomes at least 50%?" Multiple approaches to the problem are explored and compared, addressing probability concepts, problem solving, modelling assumptions, approximations (supported by Taylor series), recursion, (Excel) spreadsheets, simulation, and student preconceptions. The traditional product representation that yields the exact answer is not only tedious with a regular calculator, but did not provide insight on why the answer (23) is so much smaller than most students' predictions (typically, half of 365). A more intuitive (but slightly inexact) approach synthesized by the author focuses on the total number of "opportunities" for matched birthdays (e.g., the new "opportunities" for a match added by the kth person who enters are those that the kth person has with each of the k-1 people already there). The author followed the model of Shaughnessy (1977) in having students give predictions in advance of the exploration and these written data (as well as interview data) collected from students indicated representative multiplier or representative quotient effects, consistent with the literature on misconceptions and heuristics. Data collected from students after the traditional and "opportunities" explorations indicate that a majority of students preferred the opportunities approach, favoring the large gain in intuition over the slight loss in precision.
The CAUSE Research Group is supported in part by a member initiative grant from the American Statistical Association’s Section on Statistics and Data Science Education