# Why is a rare but unique kind of event seen as more surprising as a rare but common kind of event?

If the question sounds confusing, allow me to illustrate two examples of events.

First example: You think of a number between 1 and 1000. 1000 different people all guess it. One of them gets it right.

Second example: You think of a number. John guesses this number. He is correct.

The first example seems less surprising than the second. The second seems way more surprising since there is only one chance. But notice that this is only because when considering the first example, our mind thinks of the probability that ONE of them will guess it right. Note in the second case, our mind thinks of the probability that John gets it right. Of course the second is much lower.

The reason why we are less surprised in the first case, is because given enough attempts, a meaningful event will arise. But notice that we can generalize the second event into a more general kind as well. In the history of the world, there have probably been many times someone or a group has attempted to guess something, expect something, all with different probabilities, many of which could have even been below a probability of 1/1000. It is then unsurprising that atleast one of them would turn out to be meaningful.

What really is the difference in logic between them? In the first case, each guess is independent. In the second case, each "expectation like" event is also independent. Sure, the events in the first case might be more similar to each other than the second, but because they are independent, does it really matter? In either case, we have no prior or independent evidence that mind reading powers are possible. Is it really more likely that the person in the second case is more likely to have mind reading powers than the first? I don't think so.

And yet, one feels clearly more surprising. Is this a glitch in our brain?

• I barely understand probability so I won't attempt the math, but I can tell you that the probability of example 1 is calculable and is something far greater than 1/1000. mathwarehouse.com/probability/independent-events/… Commented Dec 30, 2022 at 16:54
• And how is this different from your previous question? Commented Dec 30, 2022 at 17:08
• In that one, one person was making many guesses. In this case, different people are making different guesses
– user62907
Commented Dec 30, 2022 at 17:12
• @thinkingman Following conifold here, this seems to be pretty much the same as your previous question, maybe we shouldn't beat that dead horse anymore? Either we tackle these questions the "physicist way" as I explained in the previous question, however uncomfortable it feels to the layman, or we move the debate to speculative territory about "glitches in the brain" and feelings of surprise. I would suggest we don't pursue the latter. Commented Dec 30, 2022 at 17:25
• Rare but common? Commented Dec 30, 2022 at 17:36

## 2 Answers

Your question is based on a misunderstanding. If you take an individual event it might have a low probability p. If you take a basket of n events each with a probability p, the chance that at least one of them will occur is 1 minus p to the power of n, which will approach 1 as n becomes very large. There is nothing mysterious about it. You are confusing the two cases. If John correctly guesses a large number, that is surprising. If you lump John's guess in with a million other guesses, the chance that at least one of them might be right is very high- however, the chance that out of all of them it should be John's guess is very low.

• I mean that is the whole point of the question lol. The question is when is it appropriate to lump it in and when is it not. It seems as if there is no concrete answer to this
– user62907
Commented Dec 31, 2022 at 0:05
• @Marco Ocram, I'm gonna have to agree. The 1000 persons guessing = 1 person guessing 1000 times. Commented Dec 31, 2022 at 14:02
• @thinkingman it depends on what odds you are trying to estimate. If you want the odds of John tossing ten heads in a row then use the individual odds. If you want the odds of anyone tossing ten heads at a world coin tossing championship, use the basket method. Commented Dec 31, 2022 at 14:18
• Yes but which one is more relevant lol
– user62907
Commented Dec 31, 2022 at 16:12
• @thinkingman Again the "relevance" is not given by probability theory, which just gives you numbers. How you interpret these numbers is up to you. There is no definite answer, certainly not in probability theory/mathematics/science about how you should interpret those numbers. If you are looking for guidance about what to do in probability theory/mathematics/science, you are barking up the wrong tree. Commented Dec 31, 2022 at 17:13

There are many limits to logic. For instance, logic dictated to the English in times past that all swans were white. A trip to Australia changed the facts. The reason that there are limits to logic is that logic in practice is subject to a property called defeasbility (SEP). Some logicians in times past (a la Frege and others in Germany) tried to deny that the logical is derived from the psychological. I suppose many people with great imaginations still believe such dissolved problems. Sabine Hoffstetter has a wonderful video on the Hempel's Raven Paradox (YT) which is an example of how human thinking about reality is constrained by empiricism, which is an old problem generally conceived of as some of sort of mutual exclusivity between rational and empirical methods (SEP). I believe there is a curious, unnamed bias of intellectual know-it-alls towards rationalism even in the face of overwhelming empirical evidence. There seems to be a surprising lack of awareness of what constitutes rationality among the philosophically inclined that one might cure by reading the Oxford Handbook of Rationality (GB) which has an excellent piece by Robert Audi.

In logic, there are fallacies such as the gambler's fallacy that hint at problem calculating probability and are specious, that is, they are reasoning that doesn't meet some general criteria such as relevance; for contemporary analytical philosophers who accept a naturalized epistemology (SEP), fallacies of logic are related to psychological claims (for some peculiar reason, some people still follow Frege down the rabbit hole of anti-psychologism (SEP)...), particularly those of cognitive science (SEP) in the area of cognitive biases. And where do many psychologists place the source of the development of such biases? In the evolutionary theory. In fact, those philosophers who embrace that have carved themselves out a neat little niche in the literature that is addressed by Stanford's article: Evolutionary Epistemology (SEP). As far as I know, there are no other contemporaneous philosophical theories in analytic philosophy (IEP), so among professional philosophers with a scientific bent, I suspect this is the received answer.

Nobel Prize winner Daniel Kahneman's Thinking, Fast and Slow (GB) is an excellent technical but lay introduction into cognitive distortions and biases.

• It sounds funny to me to say that "There are many limits to logic", as if logic should be expected to do much. What I mean by that is that "logic" of the deductive kind is just a tool which lays out rules of reasoning to avoid inconsistencies (contradictions). There is also "inductive logic" which properly used should make only very narrow claims. But "logic" in itself is not really something that will deliver deep "truths" about the world in any way. Commented Dec 30, 2022 at 21:00
• @Frank As to me. But this is a contemporaneous view. For thousands of years, pedants with bullying tendencies have attempted to constrict all valid thought to deductive logic and the classic Aristotelian syllogism as well as Euclid's axiomatic method. In the history of AI, researchers spent a few decades trying to make that happen before being heartily disabused (and IMHO in the spirit of Hubert Dreyfus, embarrassingly so). But the logically and mathematically inclined are generally but perennially, as David Dunning calls them, confident idiots.
– J D
Commented Dec 30, 2022 at 21:03
• The Medieval Scholastics with their Aristotle worship standing in for their God worship were very influential at peddling this thinking, btw. But you'll find my bone to pick in this arena extends back to Plato worship 2,500 years ago. ; )
– J D
Commented Dec 30, 2022 at 21:05
• Yes, yes. My take is that Medieval Scholastics found a "hammer" in Aristotelian logic, and tried to use it repeatedly to prove the existence of god and other doctrines of the church. They didn't know better, not having much interest in physical evidence, given they turned their heads to heaven rather than nature. Commented Dec 30, 2022 at 22:16