Carl Sagan supposedly once said that randomness is clumpy. Those three words have become one of my favorite go-to quotes, particularly when teaching residents and medical students who are often overly impressed with improbable runs of similar diagnoses or exam findings. I love this quote because it is so simple and yet reveals so much about our experience with observing the natural world. Sagan’s ability to offer up insightful nuggets of rational thought, even if he didn’t actually produce this gem, was unmatched and his efforts to bring science and reason to the public have been sorely missed. If you haven’t read any of Sagan’s works, I highly recommend The Demon-Haunted World: Science as a Candle in the Dark.
If you have a coin, and a few hours to kill, record the results of a long run of flips and you’ll see what Sagan meant about the nature of randomness. You will inevitably observe clusters of heads or tails that might seem improbable, but eventually the outcomes will average out to about half of the flips being heads and half resulting in tails. The more trials that you perform, the closer the outcomes will approach 50% for each possible result, assuming you aren’t gaming the system by using a trick coin.
I don’t think that very many people would argue with the fact that on average a coin flip is random chance, although there are still people out there who think that the Earth is flat and that Justin Bieber is a reptilian humanoid. But because of a deeply rooted cognitive bias, the gambler’s fallacy, we frequently fail at recognizing that randomness is clumpy. We accept the established overall odds, but our acceptance wavers in the face of short runs that go against our expectations. This error in logic can lead to the belief, for instance, that after five heads in a row there is a higher than 50% chance that the next flip with land on tails as if to magically even things out.
In my line of work as a pediatric hospitalist, I frequently experience other healthcare professionals making this mistake in a variety of circumstances. There is a known likelihood of bacteremia when an infant less than 28 days of life is evaluated for fever, for example. Despite this, it is common for physicians and nurses to lament, upon seeing fever as the triage chief complaint, that they are due for this life threatening infection after a number of recent febrile neonates have had negative blood cultures.
The cognitive bias which results in this commonly employed logical fallacy is, as is often the case, the result of an inappropriately employed mental shortcut. These shortcuts, known as heuristics, can be very helpful but sacrifice accuracy for efficiency of thought. In the case of the gambler’s fallacy, the representative heuristic is to blame. If someone is aware of the fact that a result has a known frequency of occurring, they often mistakenly make the assumption that short runs will be representative of long runs. They believe that a run of ten or twenty should be equally split between heads and tails in the same way that a run of a million would be. But, once again, randomness is clumpy and short runs often have surprisingly unbalanced results.
A concept that is very closely related to this is the clustering illusion. Humans excel at failing to appreciate how variable a small sample of a larger population of random or almost random events can be. We see a streak or a cluster, rather than a random clump, and assign more meaning to them than they deserve. This phenomenon can be seen at play in sports when we determine that a player has a “hot hand” but more notoriously in bogus disease clusters.
Cries of cancer clusters are common in the media. Cancer Alley, a stretch of the mighty Mississippi between Baton Rouge and New Orleans, is one such cluster that achieved widespread coverage in the late 1980’s but has largely been shown to be an illusion. There is even a Wikipedia page specifically on cancer clusters.
I recently had a somewhat heated exchange with a relative of a friend on Facebook. My friend, a mother of three boys, was expecting her fourth child and had not yet found out whether this baby was a boy or a girl. She expressed her desire for a girl and her uncle commented that the new baby would almost certainly be female because the odds were highly in favor of such an outcome. While it is true that the odds of having 4 boys in a row are somewhat low at 1 in 16, this was a classic example of the gambler’s fallacy. I responded, an argument ensued, and nobody went home happy.
So what were the odds of my friend’s child being a girl? There are two ways to approach this problem with one of them being right and one feeling right to many people because of the representative heuristic. Readers of Science-Based Medicine should of course know that what feels right on a gut level is often completely wrong. First though, some basics on the determination of sex in humans are in order.
Unlike alligators, the sexes of which are determined by the temperature an egg is exposed to at a critical period of development, whether a human infant is born as a male or female is determined by genetics. The sex of most mammals, humans included, is determined by an XX/XY system that most of you are probably fairly familiar with even if you don’t remember all the specifics.
Modern humans, individuals with genetic syndromes aside, have a genome which consists of 23 paired chromosomes. The two that determine an individual’s sex are, not surprisingly, called sex chromosomes. Females generally have two X chromosomes (XX) and males have both an X and a Y chromosome (XY). It is widely considered that human zygotes are inherently on the path towards being female at conception and that, if present, a single gene located on the Y chromosome alters this course resulting in male offspring…usually. Sex determination is very complex and there are certainly instance where the genotype (XX or XY) doesn’t match the phenotype (outward appearance) but these are quite rare and beyond the scope of this post.
Most cells in the human body are identified as diploid, which means that they contain the above-mentioned 23 pairs of chromosomes. Well, this isn’t entirely accurate, as most cells in and on the human body are bacterial and they can have up to 4 chromosome copies depending on the growth conditions at the time. But we’ll keep moving.
Reproductive cells like sperm and ova, known as gametes, are haploid in that they only contain one set of the 23 human chromosomes. This makes sense because they will combine to form a diploid zygote at conception. The female ovum always contains an X chromosome. It is the male sperm which ultimately will determine sex because an individual sperm can carry an X or a Y chromosome. Which, if any, sperm fertilizes the impatiently waiting ovum is a crapshoot and it works out to a roughly 50/50 split between male and female embryos.
Of course there is some nuance to this. There is the possibility of a minor influence by environmental factors, or factors inherent to sperm carrying X versus Y chromosomes, which may lead to a slightly increased chance of male versus female offspring in some women, or a slightly higher rate of male or female births across some populations, but these differences are not meaningful. And unless you are making use of gender selection via technology, such as with IVF, the myriad other means of encouraging the birth of a preferred sex will not alter the outcome. And studies looking at large numbers of families have shown conclusively that even in the case of families with long runs of male or female children, the chance that a subsequent child will be male or female remains pretty close to 50/50.
So in the case of my friend with 3 boys and a baby on the way, the likelihood of having another boy was 50%, not 6.25%. And the chance of finally having a girl was 50%. But let’s further explore the notion so strongly argued by my friend’s relative, that the sex of previous children impacts the sex of future children. As I have already explained, there is a perfectly reasonable cognitive bias to blame for this fallacious logic, the misuse of the representative heuristic. But for argument’s sake let’s assume that he was right.
What would the mechanism for this phenomenon be? How would past results impact future results of a seemingly random process like sex determination? Could the male “apparatus” somehow be affected by the owner being cognizant of the sex of his mate’s prior children, thus initiating a physiologic process that selects an X- or Y-chromosome-carrying champion to breach the defenses of the female genital tract and fertilize the ovum in a more controlled fashion? Or could the female genital tract somehow alter conditions such that an X or Y chromosome might have a higher chance of success? None of this makes sense.
Or maybe something supernatural is going on? We should keep an open mind, right? If you believe in a higher power then I guess you have your answer. Perhaps there is a mysterious effort underway by a technologically advanced observing alien race to manipulate the human species but still attempt to maintain at least the appearance of randomness over multiple pregnancies? Or is this some diabolical scheme where sentient sperm must act to prevent our awareness of a grand conspiracy which only works if there is a roughly equal number of male and female offspring? Are they psychic and able to probe the inner recesses of the male mind or simply plugged into the nervous system via nanotechnology? We may never know the answer but there is something I do know: don’t anthropomorphize gametes, they hate that!
Oh, and it was a girl.
One last interesting tidbit. Right now the world is on the edge of its seat waiting to see what the sex of Prince William and Duchess Kate’s baby will be. There actually is some evidence to support that it will most likely be a little girl. Morning sickness seems to serve as a predictor for having a female infant, but not just any morning sickness. A study out of the department of epidemiology at the University of Washington found that women with severe hyperemesis gravidarum that required hospitalization were 50% more likely to have a girl. But obviously this would only be helpful after the sex has been determined.