# The empirical law of large numbers and the hospital problem: systematic investigation of the impact of multiple task and person characteristics

## Abstract

The empirical law of large numbers is an important content in secondary school mathematics. Tasks used to analyze students’ understanding of this law are often based on the hospital problem, but vary in various features, leading to mixed and conflicting empirical results. To identify task features that support students when approaching this type of task, we systematically investigated the impact of multiple task and person characteristics on the accuracy of students’ responses in a cross-sectional study with *N* = 242 mathematics teacher education students. Students answered several variants of the hospital problem in different sequences. Our assumption was that differences in performance between tasks could be traced back to the salience of relevant task features and the sequence of the tasks. Results of GLMM analyses of our data support that in particular larger deviations from the expected relative frequency and a bigger ratio between the large and small sample size increase solution rates. Moreover, a verbal presentation of a 100% frequency in the case of maximal deviation increased solution rates. A within-subject analysis revealed that effects of task characteristics were more pronounced for the first task and weakened substantially for subsequent tasks. Finally, we found that 100% frequency tasks have a positive cueing effect, supporting students to solve subsequent tasks, even if the relevant features are less salient there. These tasks thus seem to be a promising starting point to connect the empirical law of large numbers with students’ prior intuitions.

## Keywords

Empirical law of large numbers Hospital problem Task characteristics Person characteristics Task sequences## References

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