The power of the complement
A tactic to handle probabilities is also a surprising trick to make sense of dubious statistics and combat fear of missing out
Probability is a tricky, unintuitive concept, that has a capacity to trip up even people who might generally be expected to know better. Imagine an urn with 10 balls in it, nine worthless wooden ones and a golden one. If you are allowed to draw one ball, you have a 1-in-10 chance of finding the valuable one. If you draw two balls from the urn, you double your chance to 2-in-10, if you pick nine, you have a 90% chance of picking the golden ball and if you pick all ten, then, well, you are certain to get the prize. Probabilities simply add together, don’t they?
A chance mistake
Well, not quite. Andrew Huberman, a neuroscientist at Stanford University and host of the Huberman Lab podcast, explains in one of the episodes why couples trying to conceive should wait at least six months before starting to worry that something may be amiss: “For women under 30, because the probability of getting pregnant on any one attempt is 20%, if it doesn’t occur the first time she should simply repeat at least five and probably six times, before deciding that there is something going on, because 20 times five is 100, so we’re talking about cumulative percent, 20-40-60-80-100, and the sixth month would take you to 120%, which is a different thing altogether.”
Different altogether, indeed. That is what happens when two intuitions clash: we “know” that probabilities can be added, and we also know that a probability of 120% is not possible. Here is how we might work it out correctly. Getting pregnant during, say, the third cycle is a combination of successive events that all need to happen: not getting pregnant during cycles 1 and 2 (you cannot get pregnant more than once at a time), and conceiving during the third cycle. To calculate the chance of these three events happening in a row we multiply the chances of the individual events: 80% x 80% x 20%, which is 12.8%. If we do this for all five cycles, we find the probabilities of getting pregnant as respectively 20%, 16% (80% x 20%), 12.8% as already calculated, and 10.24% and 8.19% for the final two cycles. The total probability is the sum: 67.23%.
If this looks a little convoluted, then that is because it is: we must work out the probability of five different sequences of events. But there is a clever trick we can use to make our calculation a lot easier. Getting pregnant and not getting pregnant are complementary events: either there is a pregnancy or there is not, and therefore the chance of one is 100% minus the chance of the other. So, the chance of getting pregnant in any of the five cycles is 100% minus the chance of failing to get pregnant in any of them. There is only one sequence of events that corresponds with not getting pregnant over five cycles: not getting pregnant five times in a row. The chance of this is 80% x 80% x 80% x 80% x 80%, or 32.77%. The chance of getting pregnant is the complement: 100% - 32.77% = 67.23%. This simple calculation produces exactly the same result as the complicated calculation of the chances of getting pregnant at every one occasion.
I have always found this a particularly powerful insight, ever since I was introduced to it in secondary school. While I still do not find probability calculus particularly intuitive, the simplicity and elegance of this tactic has helped me out many times.
Complementary framing
Yet preventing us from making Dr Huberman’s mistake, and allowing us to find the correct solution with minimal effort is not the only way in which taking a complementary view can enlighten us. As Amos Tversky and Daniel Kahneman showed in their classic paper on the Framing of decisions and the psychology of choice, it can also help us take a different perspective on the same data. In one of their thought experiments, known as the Asian disease, the US is preparing for an imminent epidemic, expected to kill 600 people if nothing is done. Two sets of two interventions to combat the disease are presented. One group of participants can choose between programme A that will save 200 people, and programme B with a 1/3 chance that 600 people will be saved, and a 2/3 chance that nobody will be saved. A second participant group could choose between programme C: 400 people will die, or for option D, in which there is a 1/3 probability that nobody will die, and a 2/3 chance that 600 people will die. You may have noticed that options A and C, and B and D are identical, but framed as complements – in the first case, survival is emphasized, in the second one it’s the deaths. Tversky and Kahneman ran this experiment with a wide range of participants, including physicians, and found overall that in the first group, 72% opted for A, and 28% for B. In contrast to what the identical outcomes of options A and C, and B and D, might suggest, the second group chose very differently: just 22% opted for C, with the certainty (that 400 people would die), while 78% opted for programme D.
This illustrates how complementary framing can lead to considerably different interpretations of essentially exactly the same outcomes. Choosing D over C signals a risk-taking attitude (typically occurring when the outcomes are framed as losses): the certain death of 400 people is less acceptable than the 2/3 chance that 600 will die. Preferring A to B (with outcomes presented as gains) tend to lead to risk aversion. While this was just a thought experiment, imagine you are facing the choice whether or not to undergo an operation, and how you would respond if it was presented as having a 95% survival rate, and if it was presented as having a 5% chance of not waking up.
And there is more. Health risks related to one’s diet are often expressed as a percentage increase of getting a health problem from a percentage increased consumption. Professor David Spiegelhalter, an expert in the public understanding of risk at Cambridge University, discusses a study claiming that “there is no safe level of alcohol intake” – any amount consumed will increase the risk of 23 alcohol-related health problems, from 0.5% (for one drink a day, to 7% for two, and as much as 37% for five). A 37% increase does indeed sound worrying. But what does that actually mean? The risk that teetotallers experience of the 23 diseases is 914 in 100,000, or 0.914%. A 37% increase in the chance of these diseases corresponds with a rise from 914 to 1252, or 1.252%. But looking at the complement – the people who do not develop alcohol-related conditions – this means that their chance of remaining free from these health issues reduces from 99.086% to 98.748%. The 37% relative increase of the chance developing one of the diseases tells a rather more alarmist story than the relative decrease in the chance of remaining free from them of 1.17%...
You can even combat FOMO using complementary framing. If you feel the urge to subscribe to ever more interesting newsletters, visit lots of exotic places, pile up the books, or keep track on social media of all that goes on the life of your friends, and feel anxious that you’re always missing out, complementary thinking can help you adopt a more stoic attitude. Don’t look at how much you are “not missing”, but at the vast complement that you are missing, and forever will be missing – the newsletters you are not even aware off, the places you will never travel to and so on. Any additional experience may feel like a significant gain (until the urge strikes again!). But if you realize that it will barely make a dent in the totality of what you will be “missing” forever more, you can interrupt the fixation and the feverish pursuit that FOMO generates. The Stoics called this negative visualization: by appreciating all that you’ll inevitably miss out on, you can let go of the anxiety over any single opportunity, and be more appreciative of what you do, and have, obtained.
Unleash the power of the complement!