Does Correlation "Sometimes" Imply Causality? I am an MBA student taking courses in statistics.
I am sure that at some point, we have all heard the famous expression - "Correlation Does Not Imply Causality".
When we are being introduced to this concept, we are often shown an absurd example of two variables that are highly unlikely to have a causal relationship, yet seemingly appear to be correlated with one another. For example - it is highly unlikely that there might be a causal relationship between the "letters in the words of spelling bees" and the "number of people killed every year by spiders":

However, there are other examples in which it might be reasonable to assume that correlation does in fact imply causality. For example, it is quite likely that there might be some relationship between "life expectancy" and "GDP" - countries with higher GDP likely have stronger economies, stronger economies likely have stronger health sectors, countries with stronger health sectors likely have more hospitals/doctors ... using this logic, although there might be outliers and exceptions, it does not seem completely unreasonable to expect that there might be some causal correlation between income and life expectancy:

This leads me to my question - when we are told that "Correlation Does Not Imply Causality", does this statement indirectly mean that there might be some instances in which "Correlation DOES Imply Causality"?
For instance, suppose I have some data (for different countries) which shows a clear linear relationship between  "the number of doctors per 1000 people" and the "life expectancy" of those countries (i.e. people who live in countries with more doctors tend to live longer on average) - although there some philosophical arguments that suggest we might never be able to "truly" attribute causality between two concepts (e.g. perhaps tomorrow we find out that countries with more doctors have some newly discovered vitamins in their water and this is why they live longer, not actually due to more available doctors, yet this is highly unlikely) - would be fundamentally wrong and morally unreasonable to prematurely suggest that "there theoretically might exist some level of causal relationship between the number of doctors and life expectancy"?

I drew this slide a few years ago that might help

Most of the silly correlations from that website are chance.  Statistics is reasonably good at describing what can happen by chance, at least if you specify in advance the correlation you are interested in.  The correlation between doctors and life expectancy is fairly robustly not explainable by chance.
The other possibilities on the slide all show correlation that's causal in nature, but only one of them is a simple 'blue causes red'.  If you find doctors are correlated with life expectancy it could be

*

*that doctors are actually good for health

*increased life expectancy causes an increase in doctors (maybe because old people need them more?)

*both the life expectancy and the increase in doctors are caused by something else. For example, maybe rich countries have more doctors (because doctors are expensive) and have better sanitation and nutrition (because sanitation and good nutrition are expensive) and that's the explanation

*selection: you've somehow chosen a sample without the low-doctor/good-health and high-doctor/poor-health countries (it's easier to get good examples of this for negative correlation)

Deciding how much of a correlation is due to causation in the way that you're interested in and how much is due to causation in annoying and unhelpful ways is hard. The goal of modern causal inference is at least to render the question precise enough that you can say what you'd need to assume and measure to answer it.
[Update: One other possibility that's not in the slide is that two things are correlated because they're actually one thing, like the radius and area of circles, but that doesn't apply to your question]
