Dear Abby: My husband and I just had our eighth child. Another girl and I am really one disappointed woman. I suppose I should thank God that she was healthy. But, Abby, this one was supposed to have been a boy. Even the doctor told me the law of averages was in our favour 100 to one. —Disappointed WomanFrom the Roman philosopher, Cicero, through the Renaissance and right to the present day, priests, mathematicians, and scientists have devoted themselves to uncovering the laws of probability. Still, chance, risk, and odds remain mysteriously opaque to many people.
Consider the doctor who told Disappointed Woman that the odds against her eighth child being a boy were 100 to 1. Because there were only two equally likely outcomes—a girl or a boy—the actual odds were one to one.
How did the doctor get it so wrong? The answer to this deceptively simple question tells us a great deal about how people think about risk and probability.
The doctor seemed to believe that Disappointed Woman’s next child would be a boy because she had given birth to seven girls in a row. He appealed to a so-called “law of averages” to support his view.
Roulette players who bet on red after a run of black follow a similar logic. Unfortunately, a roulette wheel has no memory; each spin is independent. The probability of red is the same no matter how many times the ball previously landed on black. Similarly, the sex of seven previous children is irrelevant to the sex of the eighth.
The failure to recognise this is known as the Gambler’s Fallacy. It is also known as the Monte Carlo Fallacy because it’s the reason the casino city-state is so fabulously rich.
For simple events, estimating odds and probabilities requires only a minute’s thought. But a minute is a long time and thinking is hard. To cope with our limited cognitive capacity, we rely on strategies known as heuristics—mental shortcuts that allow us to make judgements quickly and efficiently. These rules-of-thumb are like intuitions; they allow us to function in everyday life without stopping to think through every problem from first principles.
Nobel Laureate psychologist, Daniel Kahneman, gives an example of one such heuristic; he calls it “representativeness.” According to this rule of thumb, sights, sounds, and events that are consistent with (“represent”) our usual experiences appear to be more likely than those that are rarely encountered. For example, if all you knew about someone is that she has naturally blonde hair, the representative heuristic would suggest that the person is more likely to be Scandinavian than Chinese.
Although the representative heuristic is usually helpful, it can also be misleading. To Disappointed Woman’s doctor, seven girls in a row did not represent the 50 percent-50 percent distribution of boys and girls in the general population. Relying on the representativeness heuristic, the doctor wrongly predicted that Disappointed Woman’s next baby would somehow even the odds by being a boy.
Representativeness is so compelling, it can cause a health panic. The process begins when someone observes that a workplace—a school, hospital, factory—has suffered large numbers of a particular type of cancer. The usual response to such “cancer clusters” is to search for an environmental origin: high tension wires, poor air quality, and the emanations of mobile phone towers. This strategy is rarely successful because most cancer clusters are illusory. They are simply a collection of random cases that just happened to pop up in the same place.
Expecting every workplace to have the same distribution of cancer cases as the general population is a downside of the representativeness heuristic. It produces unnecessary panic and wastes resources that could be better applied to solving real, rather than imaginary, problems.
The doctor who expects every family to have an equal number of boys and girls or the gambler who believes that a roulette wheel should always produce a similar number of red and black outcomes have similarly fallen victim to the representativeness heuristic.
Representativeness is not the only judgement heuristic. Since COVID-19 came to town, the disease has dominated the daily news. As a result, surveys conducted around the world show that people vastly overestimate the risks associated with the disease.
Kahneman attributes these overestimates to an “availability” heuristic—events widely covered in the media (plane crashes, tornadoes, pandemics) are perceived to be more common than those that pass largely ignored. It may not seem true, but you are hundreds of times more likely to die of asthma than in a plane crash.
Judging relative risks seems to be particularly difficult for politicians and health officials. In Australia, fans are advised that it is safe to attend a football match along with thousands of others. But they are warned not to touch the ball should it land near them because contaminated footballs could transmit COVID-19.
A senior Australian government health official claimed that people are more likely to win Lotto than get a blood clot from a COVID-19 vaccine. This is grossly untrue; a vaccine-related blood clot, although rare, is still eight times more likely than winning Lotto.
Understanding heuristics is crucial for anyone who plays Lotto. But before I explain why, let’s get one thing clear. Since Lotto numbers are chosen at random, short of playing every possible combination, there is no way to ensure that your numbers will be chosen. Moreover, as already mentioned, the odds of winning are vanishingly small.
There are 8,145,060 ways to choose six numbers from 45. Because of the large number of combinations, the probability of choosing all six correct numbers is 0.00000012. You don’t need to be a mathematician to realise that the probability of winning if you don’t buy a ticket is 0.00000000.
In other words, the probability of winning is pretty much the same whether you buy a ticket or not. The situation is even worse for Lotto games that require seven correct numbers to be chosen from 47. The odds against winning are 62,891,499 to 1. (You don’t have to take my word for this, just get some paper and a pencil, and write out all the combinations.).
If you are still intent on playing Lotto, then the representativeness heuristic holds the key to maximising your winnings. The advice I am about to give you is the legacy of three Polish mathematicians who were employed by Polish authorities to investigate the possible corruption of their national Lotto game.
Officials noted that, on some occasions, there were many winners while on other occasions there were none or very few. To the authorities, the large variability in the number of winners did not seem random. They worried that a group of people may have received inside advice resulting in more winners than would be expected by chance.
The authorities hired the three mathematicians to investigate. They examined several years of Lotto results and concluded that there was no corruption. Instead, what they found was cognitive heuristics at work.
Like the Polish authorities, Lotto players believe that randomly selected numbers should always appear haphazard. To them, successive numbers (1,2,3,4,5,6) or other regular sequences (10,12,14,16,18,20) do not “represent” their idea of randomness. When an orderly set of numbers is drawn, there are often no winners because no one plays such sequences.
So, here’s a way to make cognitive psychology pay off. If you must play Lotto and you want to maximise your winnings, choose sequential numbers or some other regular pattern. They have the same probability as any other sequence, but almost no one plays them. If your numbers are selected, you will maximise your prize money because you won’t have to share your winnings with anyone.
Who said psychology doesn’t pay?
