Psychic Sperm:The Gambler’s Fallacy and the Representative Heuristic…..

Carl Sagan supposedly once said that randomness is clumpy. It is my new favorite thing to say because it is so simple and it so effortlessly explains so much about our experience with the natural world. Sagan’s ability to offer up life-changing nuggets of rational thought like this was unmatched and his efforts to bring science and reason to the public have been sorely missed since his passing in December of ’96. 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 an hour to kill, record the results of a few hundred flips and you’ll see what Sagan meant about the nature of randomness. You will inevitably record clusters of heads or tails that seem improbable. Runs of heads or tails five, six, even seven straight times wouldn’t surprise me at all. 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 exactly 50% for each possible result. I’m assuming you aren’t using a trick coin of course. 

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 Miley Cyrus isn’t a robot. But because of a deeply rooted cognitive bias, we tend to forget that randomness is clumpy. We accept the established odds overall, but not in short runs of randomness whether it is a series of coin flips or, for a more “real world” example from my line of work, the incidence of bacterial meningitis in neonates with fever. We do this despite the many cold doses of reality experienced over a lifetime of allowing the past results of a random process to influence our expectations of future results of that random process. This is the essence of the Gambler’s fallacy, an error in logic that 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 will land on tails. There isn’t.  

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, such as the flip of a coin or the spin of a roulette wheel, they often mistakenly make the assumption that short runs will be representative of long runs. This means that a run of ten or twenty should be equally split between heads and tails, or red and black in the case of roulette (1), 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.   

I recently had a somewhat heated exchange on the comment section of a Facebook friend’s status update. My friend, a mother of three boys, was expecting her fourth child and had not found out yet whether this baby was a boy or a girl. She expressed her desire for a girl and a relative of my friend commented that the new baby would almost certainly be female because the odds were so highly in favor of such an outcome. It is true that the odds of having 4 boys in a row is very low, about 6%, but this was a classic example of the Gambler’s fallacy. I responded and an argument ensued.

Now I am the first to admit that I am somewhat of a drive by skeptic when it comes to Facebook. I rarely allow an opportunity for chiming in when I disagree with a comment to pass by. And I further admit that I recognize that this is probably a character flaw of mine, and that I have made more than a few people rather angry or at the very least somewhat uncomfortable when their comment section is hijacked. My wife thinks I’m as ass, and as with most things, she is almost certainly correct in her assessment. I’m working on it, but I just couldn’t pass up such a beautiful hanging curveball.

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 possibly seeming right because of the representative heuristic. Readers of this blog should know that what feels right on a gut level is often completely wrong. First though, some basics on the determination of sex in humans (2).

Human infants are generally born as either male or female and the determination of sex is based on genetics. Most mammals, humans included, select gender using an XY system that most of you are probably fairly familiar with even if you don’t remember the specifics. Modern humans, individuals with genetic syndromes aside, have a genome which consists of 23 paired chromosomes. The pair that determines an individual’s sex are, not suprisingly, 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, although there is some controversy, 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 (3).

Most cells in the human body are identified as diploid, which means that they contain the above mentioned 23 pairs of chromosomes (4). Reproductive cells like sperm and ova, known as gametes, are haploid in that they only contain one set of the 23 human chromosomes. This make 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 sperm fertilizes the impatiently waiting ovum is a crapshoot and it works out to a roughly 50/50 split between male and female embryos (5). 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 bun in the oven, 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’ve 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 be? How would past results impact future results of a seemingly random process like sex determination? Somehow the male sperm would have to be cognizant of the sex of prior children and to intentionally select an X or a Y chromosome carrying champion to breach the defenses of the female genital tract and fertilize the ovum, perhaps in an effort to maintain the appearance of randomness over multiple pregnancies. These diabolical sperm must apparently act to prevent our awareness of a grand conspiracy which hinges on there being a roughly equal number of male and female offspring. Are they psychic and able to probe the inner recesses of the male mind? We may never know the answer but there is something I do know: don’t anthropomorphize sperm, they hate that!

(1) I realize that the odds of that little white ball landing in a red or black slot isn’t exactly 50/50. The 0 and 00 green slots give the house a little edge. So does the booze.

(2) As opposed to alligators, for instance, the sex of which are impacted by temperature variations rather than genetics.

(3) Usually. Sex determination is very complex. There are certainly instance where the genotype (XX or XY) doesn’t match with the phenotype (outward appearance) but these are quite rare and beyond the scope of this post.

(4) This isn’t entirely accurate. Most cells in the human body are bacterial.

(5) There is some nuance to this naturally. 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 births than female across populations, but these differences are not meaningful. And unless you are making use of gender selection via technology, such as with IVF, any environmental changes made by families to encourage the birth of a prefered sex will not alter the outcome.


4 comments so far

  1. Pete Hristov on

    “Carl Sagan supposedly once said that randomness is clumpy”
    Would you, please, point the original Sagan’s source.

    • theredstickskeptic on

      I can’t find the original source, but have seen it attributed to him. I just found it also attributed to someone named Nikki Brown. That’s why I went with the “supposedly”. It sounds like something he would say though doesn’t it?

      • Pete Hristov on

        I’ve got Stephen J. Gould in ‘Bully for Brontosaurus’ in Chs 17 , 31.
        Also, I’ve got a little problem with probability fallacies in court decisions, would you mind giving me your opinion, if possible of course. How can I get in touch with you, Sir.

  2. Pete Hristov on
    “As Sagan eloquently pointed out, randomness is clumpy. We tend to notice clumps. They stick out as potential patterns. Apparently, although we evolved to be highly skilled at noticing clumps, we did not evolve to be very discriminating about which clumps are real and which ones are not real”

    Again, no source…unfortunately.
    Gould for sure uses words such as “clumpy” and “patterns”

    Another one:
    “Randomness is inherently streaky and clumpy”
    “Mlodinow spends a lot of time debunking the notion of “hot streaks.”

    and another one:
    “Prof. Westfall of Texas Tech. He points out that truly random events tend to be “clumpy.”

    “Such probability estimates incorporate conditions before the events. Yet after the events have already occurred, those conditions may have to be discarded because the occurrences may overturn assumptions or change future behavior.”

    Overturning assumption does not changes chances, and probability does not change on the occurrence of some specific class of random events, except where there are some consequences arising out of the occurrence itself such as satellite debris. Human brain tend to register patterns, sequences of events, clumps, thus I think the statement in WSJ “What Seems Almost Impossible Can Sometimes Be Probable” is wrong.

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