What is the probability a person sees a tree by looking out of the window?

Question

I'm familiar with the basic probability rules which problems come in the form of:

• What is the probability of getting a head next by tossing a coin? 0.5
• What is the probability of getting a 3 if a number is chosen uniformly at random from the set {1, 2, 3, 4}? 0.25

How more complicated forms of questions are calculated? for example: what is the probability a person sees a tree by looking out of the window given there are 2 trees that could be seen from that window?

There are many things to consider:

• The person's eye vision.
• Time of the day (and how many light bulbs are around in case it's after sunset)
• The direction at which he looks.
• How tall are the trees?
• Can the person pass a drug test at the given moment?
• Does he have a history of delusions?

And the list goes on which may contain infinite possibilities leading to seeing / not seeing a tree. Given we don't have a frequency of people who saw a tree by looking from that very same window, how to calculate the probability with any degrees of accuracy? How confident are we given that we are 100% confident the probability of getting 3 out of {1, 2, 3, 4} is 0.25? What if we don't have anything given in which case the question is the same as the title?

Answer 1

How to calculate the probability with any degrees of accuracy??

There is no way to compute this because the estimates that we make to perform the computation have an undefined accuracy due to lack of knowledge.

The way that it is generally tackled is that we use some simplified model and apply it to the problem. But the model is wrong and we have no way to express how wrong exactly. Still, as long as the range of error is small, or smaller than the statistical variations, then the model is good enough to apply.

See also: https://en.m.wikipedia.org/wiki/All_models_are_wrong

Answer 2

Well that's what statistics is about, right? All those variables you mentioned are unobserved and can impact the outcome, therefore we choose to encode this uncertainty about the problem as probabilities.

If you have no data, there is no way to answer the problem especially when probabilities are interpreted as relative frequencies. If you interpret probabilities as uncertainty about the problem as Bayesians do, then you can make a statistical conclusion based on that. For example in the absence of knowledge, you might assume that $Pr(tree) = 0.5$ which serves as your prior beliefs. Then whatever data you do observe, you update these beliefs and end up with some posterior beliefs. Obviously the more data you observe the better your estimates will be.

Answer 3

This is what supervised learn does, particularly so-called “classification” models (most of which make probability predictions, but “classification” is all but a euphemism for predicting the probability of a discrete outcome).

Consider a deck of cards. I draw a card and ask you to guess the card without showing it to you. You have a $1/52$ probability of guessing the right card, so little less than $2\%$ chance. If I then show you that that the card is red, you’ve ruled out half of the cards and know that it must be a diamond or a heart. Your probability of guessing the right card increased from $1/52$ when you were complexly ignorant about the card I drew to $1/26$ when I gave you some information about the card.

In machine learning or predictive modeling, those details about the cards are called features or predictors (probability some other terms, too).

How to use the available features and synthesize new ones from existing features is the special sauce of accurate predictive models.

If your example, in the absence of much information about the viewer, you might think there is a low chance of her seeing a tree. However, if you know that she looked in the direction of a tree in the middle of the daytime when she should be able to see, perhaps you would think there is a high probability of her seeing a tree. Conversely, if you knew that she looked at night without the help of a flashlight and on an overcast night that even had a new moon (so no moonlight), you might expect there to be a particularly low probability of her seeing a tree.

How to model something like this is an open question that machine learning and predictive modeling practitioners tackle every day.

Answer 4

When there are that many unknowns, generally you'd say "I don't know the probability". For example, your local book-maker will not give you odds on this event with the tree, and your local insurance agent will not sell you insurance against it.

In order to produce a proabability you could take one of at least two approaches:

• Get the person to look out of the window lots of times, at various angles, times of day, etc, record when they saw a tree and when they didn't, and come up with a frequency. Use this as a probability. You do not even necessarily need to know how many of the trials occured during the day, and how many at night. You don't know what factors affect whether a tree is seen or not. You just know that you've measured the event you're interested in.

• Consider all the variables, make lots of measurements, define a more precise model to describe what is going on, put probability distributions on each of the parameters of your model (such as "time of day", "angle of glance"), and derive a probability for the event from what your model tells you are the times/angles/etc that produce tree-sightings.

Very loosely speaking, the former approach might be used by an office manager for a question like "what is the probability that someone in my office has COVID-19?", where you really can't do a lot of careful research and modelling, but perhaps you do have access to the self-reported results of various kinds of tests, or failing that to government estimates of COVID-19 prevalance in your population as a whole.

The scond approach might be used by "scientists"[*] for a question like "what is the probability that a person with COVID-19, on walking into a crowded supermarket, will infect at least one other person?", which is the sort of thing a committed epidemiologist might try to tackle. Doesn't necessarily mean all epidemiologists would come up with the same answer, of course, since they might make decisions about what to ignore, what to include in the model, and how to model it, which means they get to a different number.

You can't generally put a probability on, "my theory of physics/biology/shopping is completely wrong and therefore everything which follows from it is bunk", since you have neither a good model nor good observed frequency for that. It's best not to think in terms of "every conceivable event has a probability, and my task is to calculate it". Rather, the actual physical world has observed events, and any probabilistic model you make of the world generates probabilities, and any relation between the two is down to whether your model is any good or not.

The reason we're 100% confident that a uniform selection from 1-4 has probability of 0.25 of giving the number 3, is that this is a mathematical theorem following immediately from the definition. We are sure of mathematical definitions. No real-world events are even described in the sentence whose truth we are sure of: it's just a straight application of the definition of "uniform discrete probability distribution". The fact that we're 100% sure of mathematics (which itself arguably is a matter of opinion, but you say you are and I believe you) doesn't help us put a number on how sure we are of optics, or the medical theory of hallucinations, or that those trees won't blow down in the night and therefore the probability of spotting them tomorrow is very different from what it is today.

Statisticians working for insurance companies, however, might actually have quite good data on the nationwide incidence of trees blowing down in the night. The reason they could plausibly care is that if you have a tree near your house, they might want to have an opinion on whether they should instruct you to remove it, or at least charge you higher premiums on your buildings insurance than they charge for buildings far from trees. So, any particular factor is subject to study, but to produce a probability you always have to decide at some point to ignore everything you haven't studied.

[*] The same ones from the dread-inducing journalistic expression, "scientists say that...", which with high probability means that all scientific theory and common-sense nuance will be omitted from the remainder of the article.

Answer 5

Consider the probabilistic subject $$ \text{prob}(H | I) $$ i.e. "the probability that $H$, given that $I$". Here $H$ and $I$ are meaningful propositions, and $I$ must be not necessarily false.

For some such pairs $(H, I)$, we cannot evaluate the subject. e.g. $$ \text{prob}(\text{dogs are risible} | \text{it will rain next Tuesday}) $$

Now consider your subject: $$ \begin{align} \text{prob}(&\text{a person sees a tree by looking out of the window} | \\ &\text{there are two trees that could be seen from that window}) \end{align} $$

You haven't supplied enough information to calculate the probability, so your subject falls into the same category as my example involving dogs and rain. That's the end of the matter.

To make some quantitative progress, we'd have to add information that gives rise to symmetries in the situation which enable us to assign probabilities. For example, let $I$ be:

1. There are two trees outside.
2. Each time Bob glances out the window, he sees tree 1 with probability 0.3, and if he sees tree 1, he sees tree 2 with probability 0.1, and if he doesn't see tree 1, he sees tree 2 with probability 0.4.
3. Bob takes two glances out the window.

and let $H$ be "Bob sees tree 1 or tree 2 or both". Then we could calculate the probability.

The business of forming probabilistic subjects that are relevant yet amenable to calculation, and evaluating them, is essentially the whole subject and art of probability!

Answer 6

There are many things to consider [...] And the list goes on which may contain infinite possibilities leading to seeing / not seeing a tree.

If you want to consider even extreme cases like someone being delusional, then as you noticed, there is an infinite number of possibilities, so the the answer is simple: the probability is zero. It is something divided by infinity, so it approaches zero.

The calculation is pretty useless because the question is too broad. That is why we usually simplify such questions limiting the scenarios to consider (for most use cases you probably can ignore delusions, etc). Not seeing a tree because of being blind is a different thing than because of living in the desert.