Does logic give us a single definitive and universal answer for comparing the odds of unlikely events?

Question

As an amateur who has interest in logic and mathematics I've been reading about the concept of different probability perceptions. I'd like to have your opinions over the subject below.

When it comes to probability assessment/comparison of two unlikely events does logic give us a single definitive and universal answer? Please take this question into account over the examples below.

The events in my question concern one astronomically unlikely and measurable hypotethical event (let's call this Event 1) and another unlikely hypotethical event for which its probability cannot be easily measured yet its nature is recognizable due to being exposed to similar ones. (lets call this Event 2, more detaled explanation follows below)

To illustrate what I mean over two examples:

Event 1 is the likelihood of having an uniform picture let's say a cat photo on a random pixel generator.

Suppose we have a random pixel generator which has 1920 x 1080 screen resolution with 24 bit colors. For each pixel on the screen we have a 1 in 10^14981179 chance of being set at the correct position to generate any image we can think of hence a cat picture. (2^24^1920 x 1080) We end up with an unfathomably low probability.

I am taking the liberty of calling Event 2 a likelihood extremely surprising for some but just usual news to many people. Take crimes for instance, unfortunately every day on the news we come across several crimes therefore we are exposed to a sample unlike Event 1. Let's take hypotethically person A is committing a serious crime i.e robbery (You can name it to increase the degree of surprisingness) and it is extremely unexpected due to its not so easily explainable nature (no obvious motivation and reason for such action, completely opposite character of the person, serious consequences etc. but note that there is nothing supernatural about the action)

For me almost anything that can occur in this world would have much higher probability than the event 1 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 can a person find Event 2 less likely than Event 1 just because his experience and belief over the person who hypotetically commits the crime?

Do logic and maths tell us that likelihood comparison of these events are subjective therefore we cannot argue about the odds?

Is Event 2 a case which is impossible to measure its probability? Or regardless of the peculiarity of the person who hypotethically commits the crime is taking into account other similar cases (i.e same type of crime ratios over a certain period) sufficient to conclude that Event 2 has higher probability without any doubt ?

Thanks for reading so far. Lastly is there any specific branch or body of work that focuses on such probability and logic topics?

Answer 1

For very simple events, like particle interactions or dice rolls, we can derive quite rigorous models. Our intuition is often misleading about these cases, because while we recognise the cat is unlikely, we struggle with the idea a specific image of static is equally unlikely. We perform a 'chunking' process, where we merge many distinct states. We should understand thermodynamics like thus, with there being many similar equilibrium states like the static, and few with special properties, like the cat.

For more complex events where the parts or interactions are beyond being precisely modelled, we need Bayesian inference, where we begin with priors, or expectations. These can come from almost anywhere, we are only required to think of the best likelihoods we can. Then we adjust them over time.

It's important to understand all probability is based on a fantasy: counterfactuals. That is, we imagine the event could happen again from the same initial conditions. For simple cases, we can mimic the starting conditions closely enough, and prove this by statistical analysis. Bayesian reasoning can be used to predict complex systems, like the intentions of other humans. We know from the Dunbar Number our neocortex evolved primarily for predicting other humans, and gives us a cognitive bias towards narrating subjectivities. See: Is the idea of a causal chain physical (or even scientific)? Hume's Problem Of Induction shows us that in truth relying on the past as a guide is not a logical choice, but one derived from experience. Popper who dismissed induction's role in science, essentially argued science is about conjecture and criticism including by experiment, which fits with the Bayesian perspective.

So. Can you model the situation unambiguously? First approach. Should ve able to agree. Is the situation to complex to model completely, and unambiguously? All the computing power on Earth is limited to the quantum states of a lump of matter about the size of a tennis ball, so anything like biology that uses a substantial fraction of the available complexity, is far beyond what we can currently model explicitly. So we need conceptual chunking, and learning from experience, which will depend on framing and our past.

Answer 2

Psychology at least tells us that most humans are extremely bad at judging probability from observation, even for frequently occuring events. There are also biases like selective perception, selective memory, confirmation bias...

It's also difficult to objectively apply maths to model real world events, as any two mathematicians or statisticans can reasonably disagree on how reality should be modeled, depending also on the purpose of such an effort.

So there are pragmatic, psychological and political limitations at least to answer questions about mathematical properties of reality.

Regarding probabilistic predictions of the future, it is not possible to define one probabilistic prediction of a single event as correct. Assume a 1% chance of rain tomorrow, this is "true" both if tomorrow it rains or does not rain.

For methods of prediction, we can measure their usefulness by comparing them to outcomes over many events, but that does not make any single prediction correct or better. A wise man might much more often predict well the future than a fool, but the fool might still be right about the next day when the wise man is wrong.

Events need not be rare for this, rarity just makes it harder to compare the methods. But single predictions remain incomparable even for frequent events.

Unpredictability is similar to rarity in a way. For predicting the final position of a ball at a roulette table, similar questions in the quality of individual predictions arise.

For your random pixel generator, it is practically impossible to ever show a specific cat picture. But if you take any random picture that your generator already has generated, the likelihood for generating that random image again is still the same as for generating your cat photo. And it was the same as for your cat picture even before. So every time your generator creates a random image, it creates something that was practically impossible to happen a moment before, according to statistics. So observing anything happening does not allow use to say it was likely to happen because it happened.

We can also trivially find events that are even less likely than your cat photo generator, by using a generator with more pixels, as an example. We can find examples that are more likely, by using a generator with fewer pixels. If we observe some rare event in our lives, then without further knowledge it would seem rational to assume we might observe it again, but again, every time we observe your generator generating a random image, we can be sure we'll never see that same image again, our knowledge of the space of possibilities is just as important as observation of events.

Answer 3

The concept of probability in the second event is subjective. In the first case, it is not. The second event suffers from the reference class problem. Depending on how you characterize the situation, the probability will be different.

It is not that you don’t know the probability of the second event. It is that the very notion of a probability given the ill defined nature and context of the problem becomes vacuous.

One can probably still phrase the question a different way: If one of these events happened tomorrow, and only one, which one would you bet on if money is on the line? Most would say the second. However, this is just a hypothetical, since we don’t actually know if one of these events happened.

Arguably though, the first event seems of a kind that is physically impossible for all practical purposes. The second seems possible. For this and only this reason I would argue it makes more sense to bet on the second one over the first IF you knew beforehand that one and only one of them occurred. If you don’t know this, it is most honest to say we simply don’t know if the second event happened but that we can reasonably be certain that the first one did not.