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Key points
- People have evolved to find meaning in seemingly random patterns, but they often find meaning in places where there isn’t any.
- When people play the “she-loves-me-she-loves-me-not” game with a daisy, they often adjust the rules midway through.
- In a series of coin tosses, people expect the heads and tails to quickly even out, but that is only the case with very large numbers of tosses.
“She loves me,” says a man in love, picking off the first petal of a daisy. “She loves me not,” he says a moment later, picking off the next petal. And so, he continues removing the petals, until the last of the flower is gone and the results are in. The daisy helped the player determine whether or not he’s loved by the girl he desires.
But how did the daisy know that?
The daisy’s trick is that it gets a little help from the man himself. People play the “she-loves-me-she-loves-me-not” game with all sorts of questions, and the answer is not always 50-50. And sure enough, if we observe someone de-petaling a daisy in this way, we will usually see them adjust the rules midway through.
“I’m not even sure this is really a daisy,” a player might say as they get halfway through a flower and start to suspect that the result might be unfavorable in the end. They then go on to find another daisy to restart the game and discard the results of the first one. This improves their chances – now it’s only one in two flowers that has to have an even number of petals.
Our internal compass knows exactly how many times we are allowed to start again to balance out the game’s probabilities. If we give ourselves a 9-in-10 chance to be loved back, we might end up “cheating” three to four times, but not more. After that, if the last daisy has bad news, we’ll probably accept the verdict.
Why do we need a daisy for this game though? Surely it would be easy enough to come up with random numbers without a flower?
Strong theorems on coin tossing
“The generation of random numbers is too important to be left to chance.”
– Robert R. Coveyou
In the paper “Strong theorems on coin tossing,” mathematician Pál Révész tells a story about a high-school class. The teacher leaves the room, and the children are told to divide into two groups.
In the first group, students have to flip a coin two hundred times and record whether a flip lands heads or tails. In the other group, students have to come up with random heads-or-tails sequences, but without using an actual coin; instead, they have to “generate” random numbers in their head.
Once all the students each have 200 lines recorded, the teacher returns and tries to guess which student belongs to which group. Most of the time, the teacher guesses quite well.
Their secret is this – for the average person, it doesn’t feel right to put down five, or even four, consecutive heads or tails. That many runs just doesn’t sound plausible. However, a statistician knows that when a uniformly random coin generates a sequence of 200, it’s very likely to have runs of six or more. Its probability is close to 97 percent.
Why do people insist on explaining random events?
Amos Tversky and Daniel Kahneman point out a rule in what they call the “belief in the law of small numbers.” They found that, if a coin comes up heads three times in a row, people will expect there to be a few more tails shortly after.
What makes a coin flip fair is that it comes up heads exactly half the time – but only in the very long run. For shorter sequences, the rule doesn’t apply, yet people expect the heads and tails to quickly even out. We want to see the coin flips balance out much sooner than they actually do. We’re surprised when we see heads come up five times in a row – even if, in real life, this isn’t uncommon at all.
Understanding statistics doesn’t come naturally to anyone. When we hear a “1 in 10,000 chance,” we might imagine a stadium filled with people and one of them being the outlier. But do we really feel the tenfold difference between a 10,000-person stadium and a 100,000-person one?
We’ve evolved to find meaning in seemingly random patterns, but we often find meaning in places where there isn’t any. Yet, we know that rare events happen, and of course, they do; in that imaginary stadium, there is someone sitting in every seat. The 10,000th seat is no exception.
If we had a “1 in 10,000” chance of something happening to us, and then that thing did happen to us, it’s hard to appreciate that it was due to pure chance alone. People want to see behind the numbers. If something doesn’t happen the way we expect, we go in search of a “more likely” explanation. At the end of the day, most things happen for a reason, and “chance” doesn't sound like a good enough reason. Yet, sometimes, it should be.
