Humans struggle with understanding exponential growth due to a cognitive bias known as *Exponential Growth Bias (EGB)*—the tendency to underestimate how quickly quantities grow over time. Studies like Wagenaar & Timmers (1979) and Stango & Zinman (2009) show that even educated individuals often misjudge scenarios involving doubling, such as compound interest or viral spread. This is because our brains are wired to think linearly, not exponentially, a mismatch rooted in evolutionary pressures where linear approximations were sufficient for survival.
Further research by Tversky & Kahneman (1974) explains that people rely on mental shortcuts (heuristics) when dealing with complex concepts. These heuristics simplify thinking but often lead to systematic errors, especially with probabilistic or nonlinear processes. As a result, exponential trends—such as pandemics, technological growth, or financial compounding—often catch people by surprise, even when the math is straightforward.
I think the proper way to compare probabilities/proportions is by odds ratios. 99:1 vs 99999:1. (So a little more than 1000x.) This also lets you talk about “doubling likelihood”, where twice as likely as 1/2=1:1 is 2:1=2/3, and twice as likely again is 4:1=4/5.
90% -> 1 error per 10
99% -> 1 error per 100
99.99% -> 1 error per 10,000
That can help to see the growth in accuracy, when the numbers start getting small (and why clocks are framed as 1 second lost per…).