There are many words that carry causal sense and are quite confusing for people willing to cross the bridge from correlation to causation. I'm reading your first paragraph and I myself can feel the confusion. Correlation, regression analysis, relation, all these words can have a causal interpretation, or not.
I like to say that causality is a new lens you put to look at something. Correlation, without anything else, is just what it says it is: a correlation. If you add assumptions, it can mean something else. And only then, words like effect turn into practical effect, or a causal effect, which is what a layman person reading would expect.
It is indeed possible to estimate causal effects with regression, as long as the causal model is identifiable. Also, you may want to check Under which assumptions a regression can be interpreted causally?. These two linked questions have good and exhaustive answers.
You can have a regression in which the dependent variable is caused by the independent variables. You can have a regression in which the dependent variable causes the independent variables. In both cases, it's possible that you can achieve reasonable predictions. You can even have a regression in which neither of the variables has a direct causal relationship and still achieve good predictions.
In some places, it's possible that you can estimate ice cream consumption based on the drowning rate, that is, you can regress one on the other. However, these variables are not a direct cause/effect of each other. They're both caused by temperature/weather, a confounding variable. Things become trickier if you really want to intervene in real life, which would make quite different if A causes B or B causes A.
So yes, in terms of being able to perform a prediction with regression, you can have a regression the way you mentioned in your question. If there is a strong statistical dependence between A and B, I can predict A with B and B with A, A~B, or B~A. It will not necessarily generalize, but you can do it.
Best Answer
In a well designed lottery system, lottery numbers should be randomly generated and it should not depending on any variables.
So, if the range of the ball is from 1 to 32, I am expecting to see it is uniformly distributed, and have nothing todo with other variables such as date.
And if we really want to guess what number will be next, we have no choice but pick a random number from 1 to 32 for each ball.
Additional notes: