"Order may arise from chaos" is a common justification against teleological arguments in favor of God or a Grand Architect. Simply put, the world having order isn't necessarily indicative of there being a creator of that order. As an example, repeating digits in a random pool of numbers are a form of order found in chaos. Tribes and bonds forming out of random groups of people are too. Throwing a die and getting the same number repeatedly is also order in chaos. This, while true, fails to take into account consistency over time and how that factor may make the notion of a God more plausable.
Consistency as a challenge
If you had a die that repeatedly rolled '6' and nothing else, it would be rational to start to question if there is a reason for that die repeatedly rolling 6. If a random pool of numbers non-stop generated repeating digits and refused to stop, is that not rational to start thinking that something is causing that? If tribes formed repeatedly among common characteristics, would we not call into question the "chaos" of the situation? To move to our grander scale, the fact that there has seemingly been consistency (or at least an over all trend) in certain aspects of the universe seems to be more than chaotic chance. For example, the fact that 2+2 continues to equal 4 no matter the situation you are in is baffling from a chance perspective. The fact that, as far as our understanding and no matter the context, there are physical constants that constantly stay the same all across the universe simply would be absurd if we are subscribing to a theory of sheer chaos. At some point, it becomes more reasonable to say that something is causing a phenomenon. If a phenomenon occurs no matter the context, it might be safe to say that there is further justification to it than pure chance. Given that the universe has been alive for awhile and 2+2 is still 4 and the physical constants seemingly have been the same, it may be worthwhile to believe that there is something causing it rather than nothing.
Linking Addition to Existence
We will assume a given axiom: We as existent beings are observing reality accurately and consistently. This is just to avoid the epistemological issues of identity and whatnot. From there, we can observe that the universe has continued to exist for quite some time. It has repeatedly done this and hasn't ceased. The chance that existence simply is occurring and reoccouring seems almost impossible given amount of time the universe has been alive. It would seem absurd for this ancient universe to simply in utter chaos sustain itself. Eventually it becomes more likely than not that the universe simply breaks down its structure in the same way that physical objects do. If you left a tomato out in the sun for a week it'd rot, however if it didn't rot we'd certainly ask the question why and maybe start to believe there is an external cause for the lack of rot. Due to the infinitely small chance that an infinite chain of phenomenon is just happening repeatedly (that existence simply is repeatedly and addition is functioning as intended repeatedly), we might want to start asking questions about if there is a cause to this. Of course, we know why 2+2 still equals four as we discovered many mathamatical properties about the universe that give the consistency needed to allow 2+2 to equal 4. Would we not say that maybe we simply haven't found the cause of the infinitely sustained universe?
Applying this logic to rules.
The concept of cause-and-effect exists. It is a general rule that something needs something behind it in cause to keep it there. A book cannot be in the air without a cause for it being there. However, what exactly is the cause of the rule of cause-and-effect? It is something that exists and thus needs something to justify it? If we are to subscribe to the fact that the rule of cause-and-effect is omniuniversal (that it applies to even the causers of our universe), then that omniuniversal rule needs a causer. Something must exist to keep that cause-and-effect rule existent. Even if cause-and-effect wasn't omniuniversal, there would be a master cause-and-effect rule that would need to exist to justify the other ones. If there isn't anything, then by its own rule, it must not exist. A rule applies to itself. Take for instance, the rule that information cannot be destroyed. That rule is information and therefore can't be destroyed. If it could be, then that rule would violate itself and therefore not be a rule. If the first law of thermodynamics allowed for the law to be violated, it wouldn't be a law. If cause-and-effect as a law didn't have a causer, it wouldn't exist as it violates its own law. Thus, at some point, the law of cause-and-effect needs a causer. Even if we go through a recursion of billions of cause-and-effect laws that are functionally identical, then we'd eventually need a master cause-and-effect law that has an uncaused causer above it as to justify the chain. Cause and effect laws can't apply truly omniuniversally due to the fact that they need an uncaused causer above it, thus a cause-and-effect law can apply to all the universe below it.
What about an exception to the law
Now lets suppose that the cause-and-effect law dictated that the only exception to the rule was itself. It didn't need something to justify it, it merely was a brute fact. This is simply apart of the law and therefore makes it totally and utterly possible for the cause-and-effect law to abide by its own rules and exist without a causer. While this solution may be clever, it does have a problem. Why is there an exception? There would need to be some external justification for that exception, thus nullifying the exceptions intent of making the law possible in being unjustifiable. However, one may say that there is an exception for that rule too. That rule that gives an exception also has an exception. Repeat forever. However, such rules, if infinitely going, would need a justification for their phenomenon. Going back to the chance issue, it is more likely that there is a terminating cause than an infinite series due to the nature of increasing improbability. It is more likely that there is a something that causes the situation than there is for an infinite series just to randomly come into play without intent. It is an anomaly that those rules would even exist infinitely just to justify that there is an exception to the rule of cause-and-effect. It is more likely that there is a brute fact at the beginning of the chain of cause-and-effect that causes cause-and-effect rather than there being an infinite set of exceptions to the rule without justification. The justification cannot be simply to allow cause-and-effect to exist as a rule in itself as that would give an intent behind the existence of these rules, thus nullifying chaos and requiring the exceptions to the cause-and-effect rules to have a causer in themselves that is capable of giving intent. The rule of cause-and-effect doesn't have jurisdiction over what is above and is merely the tool of whatever the Grand Architect is, rather that be God or something else. Thus, from this, this approach of justifying cause-and-effect with itself through exceptions is a failure.
What about infinite cause-and-effect laws?
The chance rule still applies, there is still a massive issue with the fact that there could be infinite cause and effect chains justifying each other forever. If you keep going back long enough, we must start to ask "why isn't this chain broken?". Is it reasonable to believe that we'd keep rolling meta-physical dice and keep landing on '6' without a cause? We may state the cause before it, but the entire system of repeated cause-and-effect rules needs a justification. It would be simply absurd to ignore the chain due to the individual component parts happening to be satasfied. It is utterly reasonable to see patterns and identify them, especially when they have a predictability factor to them. Patters usually have causes too and thus we may conclude there is a good chance that there may be a cause to this pattern in spite of its infinite nature.
Cause and effect needs a cause to justify its own existence due to the
nature of how the law works. The law needs to follow itself or else it
isn't a law. If there is nothing above the master cause-and-effect law
to justify it, then there is no cause-and-effect. Eventually we reach a
point in where for cause-and-effect to exist, there needs to be a brute
fact. An infinite series of cause-and-effect is simply unlikely as well
due to the nature of chance and due to the fact that such a circumstance
would need justification due to our reasoning about induction. While yes
though, the infinitely unlikely chance of infinite cause and effect DOES
EXIST, it is so infinitely small that it would simply be more likely
that there exists an uncaused causer (and all the weirdness that comes
with that) then there would be for an infinitely regressing, uncaused,
chance defying recursion of infinite magnitude.
The problem of induction is yet unsolved, but lets grant that induction is a real type of logic that does exist and is valid for the sake of argument.