# How do we compare the “remarkableness” of complex events? [closed]

How do we compare the remarkableness of events? In many cases, we use probability as a tool. For example, predicting a number between 1 and 10 from a random number generator is way less impressive than predicting a number between 1 and 50000.

But what is the probability that you will wake up tomorrow at 7:30 AM? What is the probability that you will marry a woman of your liking?

Do these events have probabilities but are simply too hard to calculate? Or do they not have probabilities at all? If not, how do we compare them?

• I could be wrong, but Sean Carroll once observed something to the effect of "We have a sample size of one". We don't have reason to think that the events which happen could ever have occurred (or not occurred) any other way. May 11 at 4:17
• For events we can't model rigorously, we probably use en.wikipedia.org/wiki/Bayesian_inference Also, see 'How improbable does an event have to be before we can say it didn't happen by chance?' philosophy.stackexchange.com/questions/94079/… May 11 at 5:24
• This is a good question. "Bayesian inference" is not a complete answer as it does not by itself explain why HHHHHHHHHH coin flips are more remarkable than HHTTTHHTHT coin flips. It is necessary to explain how and why Bayesian inference comes into play. A remarkable event should not only be unlikely, but should have another characteristic... it should be exceptionally far out on some simple scale or dimension. In the way that HHHHHHHHH is exceptionally far out on the simple scale of "number of heads." May 11 at 14:47
• How do we compare the remarkableness of questions on the topic of unlikely events? May 11 at 15:33
• I studied electrical engineering and then law. My law school buddy studied computer science and then law. We are playing poker one night and argue over how to calculate the probability of drawing a particular 5 card hand from a deck of 52 cards. I say, "It is just a formula you learn in Combinations and Permutations! I'd have to look it up!" My buddy writes a computer program that runs and gives negative probabilities! A string X = HHHHHH, or whatever combination H or T, is just one equally likely outcome of a random process that happens X times and produces either an H or T with equal odds. May 11 at 17:09

How do we compare the remarkableness of events?

This is a good question. A remarkable event should be an improbable event, according to Bayesian inference, but this is not a complete answer. It does not by itself explain why HHHHHHHHHH coin flips are more remarkable than HHTTTHHTHT coin flips. These two sequences of coin flips have the same prior probability according to Bayesian inference assuming a fair coin. (In fact HHHHHHHHHH would have a higher probability than HHTTTHHTHT because the all-heads sequence could be explained by the coin being unfair.)

A remarkable event should not only be unlikely, but should have another characteristic. It should be exceptionally far out on some simple scale or dimension we were paying attention to beforehand. HHHHHHHHHH is exceptionally far out on the simple scale of "number of heads," and that's what makes it remarkable, in combination with its low probability.

Another way to look at it is that a person maintains a statistical model of the world, consisting of a bunch of parameters that give a probability distribution over possible events. We can compare this to the statistical models in machine learning. When a person sees an event, sometimes this event causes them to update their model parameters by a lot, and sometimes this event causes them to update their model parameters by not very much. If the event causes them to update their model parameters by a lot, then the event is remarkable.

Events that cause large parameter updates will be unlikely events that would be a lot more likely if some parameter of the model was adjusted. HHHHHHHHHH is an unlikely event but it becomes a lot more likely if the model parameter representing "the coin is biased towards heads" were adjusted upwards. HHTTTHHTHT is an unlikely event but there's probably no way to tweak model parameters to make it a lot more likely, because there's no parameter for "the coin is biased towards HHTTTHHTHT" that we can tweak, and therefore it is not remarkable.

In human reasoning, an event does not have a single probability, but can be predicted using various methods, each of which may result in a different probability. Also as planning agents we can influence probabilities, such as by setting an alarm to wake up, or spending lots of effort in searching for a good fit marriage partner (in the OPs examples).

In philosophy, if a system is deterministic, each event has an objective probability of 1 or 0, it will either happen or not. In a non-deterministic universe, the probability of an event changes over time. Roughly speaking the closer to a deadline you get, the more favorable or unfavorable the system developed towards the event.

An event is judged remarkable if it was notable to start with, and if our preferred method of predicting it's probability made it seem unlikely. But typically this is fraught with fallacies as humans (without training) are prone to thinking mistakes when it comes to probabilities.

Yes, you can give probabilities to waking up at 7.30 or marrying a woman you will like. The question is whether you can do so meaningfully.

The logic for allocating probabilities is easy to summarise. You identify all the possible alternatives that could happen. You know that out of all the alternatives one must happen, so collectively the probabilities of the alternatives add up to one. You have considered only the possible alternatives, so their individual probabilities are not zero. So, suppose you have n alternatives. Their combined probabilities add up to one. If you have no rationale for saying that one is more likely than another, you should give them all the same probability, 1/n. If you have some basis for doing so, you might allocate a higher probability to some alternatives than to others, keeping in mind that they must always add to one.

So, consider the question about waking up at 7.30, which I will assume means at any time between 7.30 and 7.31. How might you model the probability of that? You might, for example, say that you usually wake up between 7.00 and 8.00, so there are sixty possible outcomes (ie waking at 7.00, at 7.01, at 7.02 etc). So a very simple model might tell you the probability is one in sixty. Or you might reflect that there have been times- say once a month- when you have woken before seven or after eight. Given that, you might refine your model by saying that you have a one in thirty chance of waking before seven or after eight, and a 29 in 30 chance of waking between seven and eight. So the chance of waking at 7.30 should be adjusted to 1/60 times 29/30. However, that is a tiny adjustment, and its effect is likely to be much less than the error in your original estimate, so why bother making it? You could also refine your estimate by considering the probability that you will die in your sleep and never wake up, but again that would make a negligible difference.

How might you model the probability of marrying a woman you like? That is much harder to pin down. One way is to say that there are three possible outcomes, namely that you marry a woman you like, that you marry a woman you don't like, or you don't marry a woman at all. You might simply say the three options are equally likely, so there is a one in three chance you will marry a woman you like. Or, you might try to define what you mean by 'a woman you like'. Suppose you define it as mining a woman you like sufficiently well that your marriage does not end in divorce. You might then allocate probabilities in a slightly different way. You might look at the national average figures for men getting married to women, and find that x percent of them do. You might look at national divorce rates and find that y percent of marriages between men and women end in divorce. On that basis, the chances of you marrying a woman you like would be x/100 times y/100.

You should appreciate two key points from the examples above. Firstly that it is possible to allocate probabilities, and secondly that there are different ways to do so, each of which might give a different result, so they are simply models or indicatives guides to whether one thing is more likely to happen than another.

A more subtle point is that the meaning of the probability is the set of assumptions you have made in calculating it. What does it mean to say that you have a one in sixty chance of waking at 7.30? It means that you have assumed there are sixty possible times at which you might wake and they are all equally probable. What does it mean to say you have a 1 in 3 probability of marrying a woman you like? It means you have assumed there are three possible outcomes and they are equally possible. What does it mean to say you have an 11 percent chance of living until you are 90? It means that you have assumed your chances are based on national figures for death rates etc etc. It might be that everyone is pre-ordained to die on a specific date, and that you are going to die tomorrow, so the probability of you living to 90 is zero. But you don't know that, so you can't take that into account. All you can do is ask what is the chance that I am pre-ordained to die after I am 90, and based on national figures it is 11%.

You ask a lot of questions about probabilities, and a common denominator to many of them is that you seem to be looking for a more absolute and meaningful way of allocating and interpreting probabilities. There isn't one. All you can do is make educated guesses. All the maths tells you is that the probabilities of all the possible outcomes must add to one- how the probabilities are allocated between the alternatives is a matter of necessarily imperfect judgement because you don't have all of the relevant information.

But what is the probability that you will wake up tomorrow at 7:30 AM? What is the probability that you will marry a woman of your liking?

Determining the probability that a specific person will 7:30 AM:

1. What prior history exists of this individuals sleep schedule? With this prior history, a probability can be calculated with an uncertainty that is related to your sample size (Number of prior recorded wake-up event). Without this information, it's just a guess without any statistics to back it up.

Probability of marrying a woman of your liking?

Dating websites are full of statistical and probability models based on a list of questions designed to determine compatibility. This is necessary because the number of marriage events for a given individual (usually 1 to 3 marriages per oerson) is too small to base a statistical analysis. The variances and uncertainties are too large to make a useful decision.

Sample set size is very important. If a person has never been married before, How can a probability be established for a "marriage" event?

Remarkableness is strongly influenced by sample size. The probability for a fair coin is a number that is valid as the sample size approaches infinity.

Regarding waking up at 7:30, a mathematical definition of "remarkableness" of a particular measurement that matches the common language definition would be:

Let A be the length of a range of measurements (e.g. 7:20 to 7:40 has a length of 20 minutes) which contains the particular measurement.

Take the ratio of:

X = the frequency of measurements that fall within the particular range of measurements of length A

Y = the average frequency of measurements that fall in any range of measurements of length A selected at random on the relevant domain.

Take the limit as A goes to 0.

If X/Y is much less than 1 in the limit as A goes to 0, the event is remarkable, else it is commonplace.