Last month, Angela Merkel addressed the German people on the importance of Covid-19 reproduction number . The former quantum chemist was clear and coherent: the reproduction number, R, is the average number of people an infected person goes on to infect. With R=1.1 German hospitals will have reached maximum capacity by October. With R=1.2 capacity is reached by July, and for R=1.3 there are no beds spare in hospitals by June.
This meticulous explanation was essential not only because of the gravity of the global situation, it was crucial to convey a sense of the nature of exponential growth. We, as humans, have a lot of trouble when it comes to understanding exponentials.
Nobel laureate and economist Daniel Kahneman notably highlights how humans are good intuitive grammarians, whilst we are very poor intuitive arithmeticians. From a young age we develop a near visceral sense of verbal violation if a word is even slightly ‘speaked’ wrong. Yet we do not have an equivalent internal calculator to flash red when, for example, we are short changed by a cashier. The customer must effortfully engage in the mental arithmetic, equally so for the trained economist or mathematician short changed on her lunch break.
Just how fallible is our numerical intuition? Let’s explore.
You are one of 23 people in a room (before social distancing measures were imposed) – what is the probability that at least two people in the room share a birthday in the calendar year?
If you haven’t heard of this ‘party trick maths question’, then take a moment to estimate an answer. What do you think?
If you’ve guessed somewhere between 5-10% you join the majority of respondents who have embarked on the path of mental arithmetic which roughly adds a one in 365 chance 22 times.
The true answer is 50%. How so?
There are two ‘tricks’ to this problem:
1. The number of combinations of pairs in the room is in fact 253, 2. Probabilities compound exponentially such that: probability of zero shared birthday = (364/365)^253. Reaching for the nearest calculator that comes to 49.95%, and therefore there is a 50.05% at least one pair blows out candles on the same day in the calendar year.
In what seems frivolous on the face of it, this little maths problem highlights something important about our number-sense. Even when we mentally engage in a quantitative estimation, we are still often betrayed by our default mental rubrics and routines. This is especially true for non-linearities. As demonstrated by the so-called ‘Birthday Paradox’ our brains automatically make a linear substitution for an exponential problem – leading us to an underestimation. Our error in judgement seems trivial in regards to birthdays, it is far less trivial in relation to the pandemic.
Where these errors in judgement plague daily life, the consequences are particularly severe in fields such as medicine. An example is as follows: 1% of women who have a mammogram in fact have breast cancer. The test itself is 90% accurate. If a woman receives a positive test result, what is the chance she in fact has cancer?
Counter to our intuition, the positive test result does not dramatically shift the likelihood from improbable to probable. It in fact only increases the likelihood from 1% to ~10%. For 1000 women tested, 10 have cancer, 9 of which receive a positive test. 990 women do not have cancer, of which 99 receive a positive result, therefore, of women who receive a positive test, only 1 in 11 will in fact be confirmed to have cancer. This model for correctly updating probability from sequential information is known as Bayesian reasoning or Bayesian inference. It is an important technique in statistics and broadly across the sciences, as we recognise our poor subjective probability judgment and the need to circumvent this fallibility.
In evolutionary terms, our brains have evolved with spoken language to the order of a hundred thousand years (10^5) and short of utilising hands as an abacus (thus a base 10 number system), engaging in more advanced numerical reasoning to grapple with complex decisions has only been a facet of relatively recent human experience say a thousand years (10^3). The great irony here is our very poor subjective sense of comparing 10 to the power of 5 to 10 to the power of 3 . Many psychological studies have demonstrated our subjective number continuum is ‘logarithmically compressed’ . This means we are good with small numbers, but tend to underestimate the jump from, say, a million to a billion to a trillion. See if the following time conversion surprises you: one million seconds pass by in 12 days. For a billion seconds you would need to wait 31.7 years. How about a trillion? A trillion seconds equates to 31,709 years. A trillion seconds ago, humans had not invented writing.
Our embryonic quantitative intuition causes all sorts of problems for the complex decision making and deductions of modern life. One of our key miscalculations is in estimating exponential growth into the future. Where study participants are instructed to estimate future prices of a product based on rates of inflation, underestimation is found to be universal across cultures and across demographics . What is fascinating is how we invariably struggle with exponentials when it comes to cognition, however other aspects of our physiology process exponentials as the default. For example, we hear sound on a decibel (logarithmic) scale. Similarly, we perceive intensity of light as logarithmic to the actual number of lumens. Intrinsic to our mental underestimation of exponentials, is our poor sense of the effect of growth factors.
In a global crisis underscored by models of exponential growth, the need to overcome our intuitive miscalculation has never been higher. We must find a means to truly internalise the importance of a growth factor. The case by case picture described by Merkel was a valiant attempt to bridge the cognitive divide by drawing the exponentiated future consequence into the present. This bridging is essential when it comes to citizens seeing the necessity in complying with restrictions and “flattening the curve”.
In any crisis, evidence-based decision making from quantitative data is paramount. In a modern society where misinformation is rife and attention bandwidth for complex data is narrow, communicating scientific information clearly to a wide audience is a challenge of its own. When the key concept which is one in which we lack natural intuition for, the challenge is twofold. We must recognise that humans have evolved as story tellers, and not as intuitive theoreticians. Therefore, the most compelling mode of communication is to convert the numbers to narratives. The story of a growth of factor of 1.3 has a vastly different ending to the story of 1.0. Rather than stating the number-value of the growth factor and its exponent, we must instead tell the story of the power with which these numbers hold.
 The Guardian – “Angela Merkel draws on science background in Covid-19 explainer” https://www.theguardian.com/world/2020/apr/16/angela-merkel-draws-on-science-background-in-covid-19-explainer-lockdown-exit
 Karolis, V., Iuculano, T., & Butterworth, B. (2011). Mapping numerical magnitudes along the right lines: Differentiating between scale and bias. Journal of Experimental Psychology: General, 140(4), 693–706. https://doi.org/10.1037/a0024255
 Keren, G. Cultural differences in the misperception of exponential growth. Perception & Psychophysics 34, 289–293 (1983). https://doi.org/10.3758/BF03202958