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Improbably Right

April 24, 2023 · Milan Lajtoš
story, thinking, cognitive bias

Sometimes you just have to be right. Even if it's improbable. Even if it's impossible. Even if it's a little bit crazy. Even if it's a little bit stupid.

Here are the three stories abouth being right.

Improbable hens

When I was 10 years old, math teacher was giving our class a flash test — word problems from exercise book that were pretty trivial – everybody solved them without too much thinking and we didn't even had to do any calculations on the side. Just write down the answer and then raise a hand to signal that we were done. When every hand was up, we moved onto another question. My hand was constantly up with the fastest ones. We moved from question to question, the pace was pretty rapid – we didn't really waited too much for anybody. And then this question came up:

3 hens lay 3 eggs in 3 days. How many eggs do 12 hens give in 12 days?

My initial reaction was that this is kinda hard to solve just in my head. So I started to scribble some numbers on the side. By this time, every hand was up but mine. I was shocked and ashamed. I asked for some more time. Everybody including the teacher laughed. My face must have been as the ripest tomato under the sun. After few more seconds I raised my hand. I was in disbelief that I was the slowest to solve it. We continued with more rounds of questions. I could not shake off the bad feeling.

After all rounds were finished, we were going one by one and checked our answers by shouting what we got. I was waiting for that one question in particular. When we got to it, everybody shouted "Twelve!". I didn't shout. Just said "48" with the most embarrassed voice in my life. I was the slowest to come up with the answer and at the end, I was the only one who got it wrong. The teacher was perplexed of why I said such wrong answer to such simple question. I never felt more ashamed of myself before. We moved on to check the rest of the answers.

After we went through all the questions, the teacher checked the exercise book for the correct answers. And to everybody's surprise there it was – 48. I was asked to explain why it was the correct answer and I did. With the biggest grin on my tomato face.

Super-sophisticated visual illustration of multiplicative relation between hens, days and number of produced eggs.

Improbable speed

When I was around 15 years old, I happend to be in front of a blackboard, answering a physics question asked by our teacher. The question was without any prior context:

How would you get an average speed of a moving car?

Without hesitation, I answered that I would measure car's speed at every instant and produce an average of those measurements. Teacher laughed and asked me how to make an infinite number of measurements. When she expressed I should use something practical, I produced the usual answer – average-speed := (distance / time).

Few years later in math class, I remembered that moment from physics class. We were just presented with integral calculus and there it was – you can get an average of some function with definitive integral. That is literaly measuring car's speed at every instant! Not very practical, but ultimately true.

Improbable logic

When I was grad student, we had intro to cognitive psychology. Every week we studied some aspect of human cognition and then we presented our findings. One week's topic was logical reasoning. I had to do some test and send my results to the person that was presenting the topic. The task was to check whether specified rule holds or is broken with presented cards. An example of what the task looked like is here. If you want to think first, don't read the text below the box.

If a card shows an even number on one side,
then its opposite side is blue.

Which cards you need to turn to check if the rule holds?

(Every card has number on one side, and blue/orange color on the other.)

Let's think through which cards we need to check.

  1. The first card has a "3" on its face. Does the rule say something about odd numbers? No, so we don't have to touch this card.

  2. If there is "8" on the second card, as the rule suggests, we need to check if the opposite side is blue, right? That's first card that needs to be checked.

  3. Next, do we need to check the third card? One would think that we would need to turn the third card which is blue, to check whether there is an even number on the other side, right? Not really. The rule doesn't say there can't be a card with an odd number and blue color. So if the card is blue, it doesn't matter wheter it has even or odd number. We don't have to check this card.

  4. Do we need to check the fourth card, which is orange? Rule doesn't say anything about orange color, right? But what if there is an even number there – that would mean, the rule is broken. So we need to turn this card as well.

Some people might recognize this task as test for material implication. At first, the test might seem counter-intuitive and not really straightforward, but it is perfectly logical.

Back to the story.

When one of my peers presented the results on the Wason selection task, only one participant scored 100%. It was me. Again, the blood rushed to my face as I thought everybody was thinking that I was a cheater. And in fact, I kinda was cheating. I know that humans are biased for intuitive understanding rather than logical one. Intuitions are shortcuts that are very valuable, but sometimes they are just plainly wrong. This was one of those shortcuts and everybody but me took it.

But wait, there is more – this task has a hidden twist. When Wason selection task is presented in context of social relations (or more broadly "evolutionarily familiar problems"), people tend to get it immediatelly, or rather intuitivelly. Bonkers, right? Here is an example of this social grounding:

If a person drinks an alcoholic drink,
then they must be over the age of 18 years old.

Which cards you need to turn to check if the rule holds?

(Every card has age as number on one side, and milk/beer icon on the other.)

Easy, right?