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I've been pondering the concept of subjectivity to evaluate likelihood and predictability of extremely rare events. I'd like to demonstrate my question over trivial hypothetical examples below. I did some research but couldn't find specific answers in the area of probability and statistics. I am hoping to have answers from the community here. The explanation part may seem a little lengthy, thanks in advance for reading.

Some incidents can be extremely surprising for limited number of people but perceived as just usual news to many people. Take crimes for instance. Likelihood of a person's commission of a serious crime i.e. robbery can be extremely unexpected (you can change type of the crime to increase degree of surprisingness/shockingness) due to its not so easily explainable nature (no obvious motivation and reason for such action, completely opposite character/behaviors of the person, serious consequences etc. thus execution of the action looks arbitrary but note that there is nothing supernatural about it). However the same action can be perceived by a stranger just as a usual case, one of the news on TV, as unfortunately every day on the news we come across several crime incidents thus we are exposed to a sample.

To illustrate this as Event 1: Let's say Person A is waiting a bus in front of a bank office. He knows that likelihood of attempting a bank robbery for him is 0%, impossible. There is no reason at all for him to do such action as he isn't an immoral person and he is neither mad nor psychopath. He's looking forward to arriving home, having dinner with his beautiful wife and lovely children etc. We can change the subject from Person A to ourselves to stress 'It won't ever happen because I know.'

Let's illustrate this time an absurdly unlikely event for which its likelihood can be straightforwardly calculable. I'm calling this Event 2:

Having a uniform real image, a cat picture for example, on the screen of a random pixel generator with 1920 x 1080 resolution and 24 bit colors is 1 in 10^14981179 chance. (2^24^(1920 x 1080)) We end up with an unfathomably low probability.

For me almost anything that can occur in this world would have much higher probability than Event 2 which is absurdly improbable. Let alone the lifetime of our universe, mathematically millions of universe wouldn't be enough to see a uniform real cat picture on a random pixel generator even it shuffles the pixels every second.

However,

Is it also logical to assess likelihood of Event 1 lower than that of Event 2? Can P(Event 1) < P(Event 2)? (by approaching robbery case in Event 1 unique and peculiar not just a statistic, no matter how absurdly low chance to have a cat picture on the random pixel generator it is also absurd for Person A to commit such crime, maybe literally 0% chance).

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  • $\begingroup$ I had to ask the question ''Can P(Event1) < P(Event2)?'' to better explain the preceding question. Sorry as well if the examples about crime part are disturbing, this is just a mathematical discussion. $\endgroup$
    – Geerts
    May 28, 2022 at 8:47
  • $\begingroup$ Please note that I'm referring having a uniform real cat picture on the screen of random pixel generator to be able to demonstrate an absurdly low probability. Static noise images that can be perceived as a cat picture are irrelevant with my question. Intention here is to be able to show a calculable absurdly low probability as a reference to compare assumed probability of Event 1. $\endgroup$
    – Geerts
    May 28, 2022 at 12:22
  • $\begingroup$ What confuses me in your description of event 2 is that you state 'a uniform real cat picture'. But what is 'uniform' and what is 'real'? You probably mean a specific single picture, one out of all $10^{14981179}$ possibilities. But this computation is distracting on a statistics forum. Besides this one possibility that you computed there are many other possibilities to arrange the pixels such that you still have a 'cat' in the picture. (It is probably not relevant for the question, which seems to be just about a very unlikely event, but it is distracting) $\endgroup$ May 28, 2022 at 12:53

2 Answers 2

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Can P(Event 1) < P(Event 2)?

Yes, we have $$P(\text{event 1}) < P(\text{event 2})$$ because $$P(\text{event 1}) = 0 \qquad \text{and} \qquad P(\text{event 2}) = 10^{14981179}$$

Is it also logical to assess likelihood of Event 1 lower than that of Event 2?

However, this problem does hinge on the assumption that the bank robbery is impossible and $P(\text{event 1}) = 0$. And this assumption is just some believe by a person and not a calculated probability.

So we should not regard the current comparison as a mathematical computation and comparison of probability.

Also, we can not make that mathematical computation unless you want to go so far as computing the probability of all the random processes in the neurons and environment falling into place to make the person A commit a bank robbery. This is practically unfeasible unless you build a computer that, instead of spitting out the number 42, is able to make a googolplex Monte Carlo simulations of earth and it's inhabitants.

So, not it is not logical. The argument fails by considering $P(\text{event 1})$ as something that can be assessed in a mathematical formula like $P(\text{event 2})$.

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  • $\begingroup$ Indeed likelihood of Event 1 is an assumption. I already indicate in my post that it isn't easy to make modelling of such events. Yet, can it be possible that assumed probability of Event 1 is lower than the astronomically low probability of Event 2? I guess many people would approach such case in Event 1 just like a statistics no matter how unexplainable it looks. I'm questioning if someone can find occurence of Event 1 absurdly unlikely by assessing it as a unique case due to its nature. Please see the explanation I made in paranthesis which follows my question at the end of the post. $\endgroup$
    – Geerts
    May 28, 2022 at 10:55
  • $\begingroup$ @Geerts maybe I do not understand your post. To me it seems that you are asking in some indirect way whether $$0 < 10^{14981179}$$ But also, you seem to have something non-mathematical in your question. This is not easy to grasp because you are trying to explain this non-mathematical issue with a mathematical equation. My answer tells, don't use mathematics to explain non-mathematical concepts, it gives a false believe of mathematical rigor to things that are not. $\endgroup$ May 28, 2022 at 11:00
  • $\begingroup$ BTW I find $10^{14981179}$ a bad expression (there are many more images that look like a cat). And it is also not a very small probability. It is a bit weird to see this as a sort of lower limit of practical probability. There are smaller probabilities. What about the probability to have an 8K randomly generated video to be the same as Coppola's Godfather trilogy? Or what about the probability that my bicycle tire with a puncture will inflate rather than deflate and as a consequence explodes? $\endgroup$ May 28, 2022 at 11:08
  • $\begingroup$ Thank you for your effort but I believe my question is precisely clear. The fact that there are distorted and static noise images that may look like a cat is irrelevant with my question. I'm clearly referring a uniform real life cat picture in my question to demonstrate a calculable absurdly low likelihood. Importance here is a calculable absurdly low likelihood to be able to compare it with the odds of Event 1. $\endgroup$
    – Geerts
    May 28, 2022 at 11:35
  • $\begingroup$ On the other hand I'm showing another case (Event 1) which has low probability due to the reasons I explain in detail in my post. Event 1 isn't straightforwardly calculable . Thus, some people may assess it just like another robbery case on the news. However due to not so easily explainable nature of it (no obvious reason, opposite behaviors etc. see the details in my post) it looks shockingly arbitrary. I'm asking instead of taking Event 1 just like a stats, can its assumed probability be lower than that of Event 2 which has absurdly small probability? $\endgroup$
    – Geerts
    May 28, 2022 at 11:36
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Your question doesn't seem to be specifically related to the rarity of the events, but to the fact that different people can have different probability assessment for the same event, based on their different knowledge.

This is called Conditional probability. For example, the probability that an arbitrary person will get cancer $P(\text{cancer})$, is different from that of a smoking person to get cancer $P(\text{cancer} | \text{smoking})$. In general $P(A) \ne P(A|B)$ for two events $A,B$ (unless they are independent). Therefore, if you ask yourself what is the probability of a certain person to get cancer (or rob a bank), the answer will be different depending on what other facts you know about them.

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  • $\begingroup$ Indeed essence of my question concerns conditional probability. (It's one of the tags of my post) Do you think than probability of a person's commision of a serious crime (e.g. bank robbery, murder etc.) can be even lower than absurdly unlikely cases like having a specific cat picture on the screen of a random pixel generator (Event 2)? Important point here is the action looks arbitrary so there is no obvious reason as mentioned in the post, thus a specific person's commision of the crime is shockingly unexpected in Event 1. Can its probability even be literally 0%? $\endgroup$
    – Geerts
    May 29, 2022 at 17:21
  • $\begingroup$ Theoretically it can be any number between zero and one, including zero. $\endgroup$
    – J. Delaney
    May 29, 2022 at 18:02

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