However science can take a position on "we know there are these things which would cause exactly this" or at least "we may not know the cause but we have repeatedly seen this happen occasionally". So science certainly can say that something is not a miracle, only a low-probability event. More than that, it must, because if you attribute anything at all, ever, to divine intervention then you destroy any possibility to investigate a physical source. And that is directly opposed to rational thinking. We have historical proof of why this is wrong.

Any valid long English text encoded as ascii or whatever you specify in advance found early in pi can be your possible message. You can limit acceptable messages so the odds are you will not find one. You might find a message in French and throw it out, and thus fail to discover something valid. Or you might accept it, and maybe discover nonsense because you have violated the confirmation bias limits. Or an English message might be valid or invalid. Or there might be nothing to discover. You can arrange so that if you find something, the odds are it isn't chance.

If you do find something, the process doesn't tell you if it is unusual luck, great wisdom, a great lie, or something else.

However science can take a position on "we know there are these things which would cause exactly this" or at least "we may not know the cause but we have repeatedly seen this happen occasionally". So science certainly can say that something is not a miracle, only a low-probability event. More than that, it must, because if you attribute anything at all, ever, to divine intervention then you destroy any possibility to investigate a physical source. And that is directly opposed to rational thinking. We have historical proof of why this is wrong.

Your problem there is confirmation bias. If you have a sufficiently large random sample (such as, hypothetically, the digits of pi), then any encoding you care to pick (maybe ASCII) will eventually find a string of digits which produce the message "I'm God and I'm a black woman. Fix the Sistine Chapel ceiling.". Just for an example. It's equally likely as the encoding producing "zadfafhtyuweljhbgjkfyfqemgeghrhdf", because that's how probability works. The risk for miracle hunting is that you find something because you're looking for something without knowing exactly what.

My point is that you have to say beforehand what message you're looking for. If you can't do that, then you can't calculate odds, because that would be asking "what's the probability of finding something I've already found?" to which the answer is "exactly 1, because you've already found it". But if you can predict it then by your definition it's not a miracle, so you'rea bit stuck there.

Edit - Comments on several answers are getting lengthy, and likely will be moved to chat soon. I have made several comments. I am moving some of them here. These are from comments on gandalf61's answer, where he says

... there is no empirical evidence for supernatural causes, for miracles, or for magic.

My comments address limits of what physics can say about biology, or perhaps what science can say about reality.

+1, but be careful about saying there is no evidence for miracles, etc. Science is all about laws of nature - repeatable patterns of behavior of the universe. A miracle is by definition not a repeatable effect of any physical cause. One could happen right there in the lab, and a good scientist would throw it out because it can't be verified by repeatable experiment. Not to say that miracles happen in the lab or elsewhere. But science has no evidence because by design science does not consider that kind of evidence or include that kind of artifact in any theory.

Science has no evidence for or against miracles. The lack of evidence is to be expected, and is not evidence for or against the existence of miracles. Science can't take a position on the question because it doesn't look for evidence.

However, science can take a position on other religious questions, such as the effectiveness of faith healing. Here a healer asks God for healing. We can measure if healing takes place and compare to similar cases where nothing is asked. I have heard of serious consideration about such a clinical trial. There are difficulties. How does one arrange a double blind trial, where neither healer nor patient knows if genuine or placebo prayer is being offered?

There are places we could look for things that have no natural cause. Messages encoded in the digits of pi. Patterns in quantum randomness. There are also areas where we have found strong patterns - natural laws - without being able to anything about their cause. I am not arguing for or against anything supernatural. Just trying to clarify the limits of science.

Graham raises some points.

... it is absolutely correct to say that we have no evidence for miracles. The main issue is the "any physical cause" part. In all experiments, the biggest problem is isolating the many other potential physical causes from the one(s) you're investigating. Every experiment already has a repeatable source of "miracles", which is the rest of the world injecting noise which gets past your experiment design.

However science can take a position on "we know there are these things which would cause exactly this" or at least "we may not know the cause but we have repeatedly seen this happen occasionally". So science certainly can say that something is not a miracle, only a low-probability event. More than that, it must, because if you attribute anything at all, ever, to divine intervention then you destroy any possibility to investigate a physical source. And that is directly opposed to rational thinking. We have historical proof of why this is wrong.