Humans are, by default, bad at probability. Let’s take the infamous Monty Hall problem to illustrate this point. It goes like this: on a certain TV show, you appear as a contestant. In front of you are three doors. Behind two are goats, while the third has a cool sports car. Of course, you don’t know which door has what behind it. You can pick one of the three and claim whatever is found there as your own. You decide to go with #1. The game show host then opens door #2, behind which is a goat. Before you open your door, he permits you to switch your choice to #3. What should you do?

The solution is simple: switching doubles your chances of winning. This is shocking to some, but is rather easily explained. Let me illustrate:

We need to calculate the probability of door 1 having the car given that door 2 has been opened to reveal a goat. This automatically rules out #5 and #6. For door 1 to contain the car, scenarios #3 or #4 need to have occurred. However, in both these cases, the host can open either door 2 or door 3, making the chances of door 2 specifically being opened half of the probability of #3 and #4 occurring. For #1 and #2, the host is forced to open door 2, effectively making their odds double that of #3 and #4, which is only possible in a 2/3–1/3 split.

This problem caused widespread uproar in academia, with even Ph.D. holders contesting the result published in Marilyn Von Savant’s Ask Marilyn column in 1990. Many outright refused to believe the proof even when supplied with one; only real-time experiments got them to agree that switching was indeed always the better choice. The one thing which this whole ordeal shows is that the human mind is patently unsuited to understanding probability, even seemingly straightforward cases like above.

Let me give another example where the flawed human perception of randomness caused several bankruptcies and allegedly even suicides. In 2003, someone noticed that the number 53 had stopped appearing in the Venice lottery. As time passed, this inspired a cult-like state in regular players: the longer it stayed undrawn, the most reckless the bets became. Many lost their lives’ savings in this mania. Some killed themselves after their failure. Eventually, after 153 draws, the number finally showed face, more than two years since its last picking.

The cause of this whole fiasco is something called The Gambler’s Fallacy. Imagine that your friend tossed a coin 99 times, and it turned up heads each time. If you had to bet on the result of the 100th toss, what would you choose? Surprisingly, a majority will pick tails: the reasoning is that since heads came up so many times, sooner or later, it would have to be tails’ ‘turn’. True randomness has no memory and does not care about history, so if the coin is fair, then the chances of tails on the 100th toss are still 50–50. That said, I would probably bet heads since it more likely than not that the coin is weighted.

Not only are humans terrible at understanding randomness, but when we do take it into account, it is usually overdone. Here’s a trick you can play at your next family gathering: get two people and give one a coin. Ask them to flip it 30 times and record the results. The other person has to write what they think would be a possible outcome of a coin being flipped 30 times. If you examine both sheets of paper afterward, you can quickly tell which was which: the real one will have a longer run of the same side (so for example, five heads in a row) and fewer switches, i.e., opposite outcomes between successive flips. Humans tend to keep a ‘mental ledger’, and are not very likely to include a long series of the same side. In reality, it is more likely than not that in 30 flips, there is at least one run of 5. Likewise, humans like to have roughly equal numbers of tails and heads, despite that being statistically improbable.

The reason humans suck at understanding chance is that our brains aren’t suited to it. We are made to understand macroscopic cause and effect – which doubtless helped our ancestors in hunting and agriculture – and are somewhat good at it. Randomness, on the other hand, is… random. There is no causal relationship to be found here; only pure mathematics can be relied upon. This is precisely the reason why probability was one of the last of the core branches of math to be developed: our ancestors believed chance had an unknown, divine cause, which kind of shut down any possibility of discussion. The origin of the now-common practice of using a coin flip for decision-making also derives from such superstitions. Roman citizens used to flip coins with Julius Caesar’s face on them to end conflicts; if it landed on heads, it meant that Caesar agreed with the flipper. I am still unsure exactly what happened if both arguers got tails.

There have been some serious miscarriages of justice due to mistaken probability, too. One such flawed use of logic is the Prosecutor’s Fallacy. Take the case of Sally Clark, an English solicitor wrongfully imprisoned for the deaths of her two children, both of whom died due to Sudden Infant Death Syndrome (SIDS). The prosecutor argued that since the chance of a single cot death was around 1/8543, the value for double SIDS cases would be 1/8543², or one in 73 million. Based on this, Sally was deemed guilty and incarcerated. In reality, there were more things wrong with this figure than not, some of which were:

1. The wrong probability was considered. It makes no sense to take the arbitrary value of 1/8543 and square it. That is like saying that since there are seven murders per 100,000 people per year, any given case is overwhelmingly not a murder. There is no reason to include live people in the statistics; just like in Sally’s case, there was no reason to include all the live births. The correct probability to consider would be the chance of foul play in families with two dead infants.
2. Even taking the 1 in 73 million figure at face value does not lead to the conclusion that Sally was guilty. Around 140 million babies are born each year, so it makes sense for there to be someone who lost two kids to SIDS. One thing is clear, though: double SIDS is more likely than double infant murder.
3. Finally, the 1/8543 figure itself was erroneously produced. It failed to consider the fact that both babies being boys increased cot death chances and neglected genetic effects.

None of these three mistakes were caught at the original trial. Sally was imprisoned for three years and had her public image reviled before her subsequent reappeal. Even after her release, the grief of her children’s demise and the injustice of the false conviction weighed heavily on her. In prison, she was treated harshly due to her reputation as a child-killer. She developed Alcohol Dependency Syndrome and various other psychological conditions, unable to come to terms with the past few years. She had no quality of life and died in 2007, just four years later, of acute alcohol poisoning.

This shows that humans tend to over-rely on our intuitions for understanding things not suited for it to our detriment. Sally Clark was not the first to be unfairly accused of something they didn’t do on the basis of flawed statistics, and she won’t be the last. Until then, all we can do is to ensure we don’t ourselves fall into the many traps of chance and help others avoid them as well.

## The Calkin-Wilf Tree

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