Erroneous estimation of the factorial function and improvement through calibration experiences

Abstract

Factorials are important for counting collections in discrete probability and combinatorics. However, people have difficulty reasoning about them. Tversky and Kahneman (1973) found that under time pressure, people massively underestimate the expansion of $8!$ (correct value $40,320$), and that the degree of their underestimation is less when the product is presented in descending order ($8\times7\times6\times5\times4\times3\times2\times1$; Median = $2,250$) vs. ascending order ($1\times2\times3\times4\times5\times6\times7\times8$; Median = $512$). We attempted to replicate both findings in a sample of N = 140 participants and to assess whether calibration reduces underestimation errors. Participants first estimated both orders (counterbalanced). We replicated the massive underestimation on the first attempt but failed to replicate the effect of order between-subjects. However, participants significantly improved their estimates when answering the descending order after the ascending order, a within-subjects effect not previously demonstrated. Participants were then calibrated to the correct value for either $6!$ or $10!$ and estimated both orders of $8!$ a second time. Participants who received the larger calibration value ($10!$) made much more accurate estimates for $8!$ (Median = $38,000$), which in fact did not differ statistically from the correct value. Participants who received the smaller calibration ($6!$) still grossly underestimated $8!$ (Median = $2,678.5$), despite $8!$ being closer to $6!$ than $10!$ in linear and log units. Our findings suggest that people’s underestimation may be (1) lessened by the presentation of descending after ascending order, and (2) greatly reduced by providing an upper-bound reference ($10!$). This may have implications for mathematics and computer science instruction.