Affirming the consequent
Scientists distinguish between the merit of explanations on the basis of (a) how accurately and (b) how widely they make experimentally-verified predictions.
This means empiricism is fundamentally based on affirming the consequent (and uses inductive reasoning), so you could argue empiricism is rather weak logically. However, you should note that having used
A => B and
B, science does not assert
A is true, it labels
A a good model for
Sadly, the media and popular opinion tend to conclude "scientists discovered that A is true". This common fallacy ("science is all true") leads to disillusionment when more data suggests a different conclusion.
Science is fundamentally not about truth, it is fundamentally about degree of statistical agreement.
This is why Newton is held as an eminent scientist, despite his theory of mechanics being demonstrably incorrect. Science doesn't care about correctness, it cares about degree of accuracy of prediction, and Newton was very accurate.
Yes your derivation is correct. I would have presented (5) as:
P(H|E)=1, so by Bayes' theorem P(E|H) = P(H|E)P(E)/P(H), we have
P(E|H) = P(E)/P(H), but similarly, P(H|C)=x<1, so
P(C|H) = x P(C)/P(H), but P(C)=P(E) by prior assumption, so
P(C|H) = x P(E)/P(H) < P(E)/P(H) = P(E|H). Thus we have shown
P(C|H) < P(E|H).
but yours has the merit of being (a) briefer and (b) more explanatory.
Models that are validated empirically
This argument from Bayes' theorem explains why simple explanations making firm predictions are validated over ones which make more equivocal predictions, given similar accuracy. More generally, empiricism's statistical approach favours theories which readily make (preferably numerical) specific and universal deterministic predictions from measurable inputs over theories which make non-deterministic predictions or where it is complicated to find what the theory predicts.
This is absolutely the right approach to take in a scientific paper, since peer-reviewed empiricism is the validation tool for science - not logic, not "the truth", but statistical predictive accuracy.
That's why "children prefer to leave their toys messy than tidy" isn't part of any major scientific model, whereas "the Higgs boson exists" is. The empirical evidence for the toys is overwhelming but hard to quantify and only nearly universal, whereas the evidence for the Higgs Boson is very localised and of comparatively incredibly rare frequency, but easy to analyse numerically.
Thus another too-common belief "all truth is science" is also fallacy.
It is one which, amusingly, would tend to be confirmed by empirical study (because non-numerical, multiple-option and generally hard-to-statistically-analyse truths already evaded being part of the scientific model)!
Can we use this method to decide between theories?
You have a lovely statistical argument in favour of a scientific theory (evolution) over a competing theory (young earth creationism). However, as always with science and statistics, you should be cautious about overgeneralising or misstating your conclusions.
It should come as no surprise that a statistical analysis favours a deterministic theory, as we just proved using no assumption other than E->H and C-/->H that P(E|H)>P(C|H). Don't allow your belief in evolution to encourage you to accept this kind of reasoning as a proof, since you'd be making the fallacy of affirming the consequent, about affirming the consequent!
To dissuade you, I'll give an example the other way around. I could compare evidence E of non-zero credit card bill with theory G: "the nasty green goblins will always cause the numbers to be non-zero on my credit card bill" and theory C:"I prefer to put purchases made at non-small businesses on my credit card to defer payment with "cashback" deducted, maintain a higher current account balance generally, and put the resulting spare money in an interest-earning savings account". Every month, using Bayes' theorem, P(E|C)<1 (albeit slightly) and P(move money into savings)<1 (significantly less slightly!), but P(E|G)=1, so I find that after a year or two, P(G|EEEEEEEEEEEE)>P(C|EEEEEEEEEEEE) by a wide enough margin to be statistically significant, so to behave empirically, I should accept the goblins model.
What can we conclude?
Don't let the green goblins near your credit card.
No, no, that's not it!
To reason this way has logical gaps (as you spotted, plus general inductive reasoning including assumptions about the immutability and uniformity of the universe), favours certain kinds of statements over others (Higgs vs toys), which can lead to misdiagnosis of the truth (credit card).
However, it's perfectly valid to use empirical statistical reasoning to decide between two scientific models. This way you deliberately favour the most statistically predictive theory, but remember, that means you have a good model, not necessarily the truth. (Your school education should have exposed you to a good few models that were the best explanation we had but were later surpassed. Be particularly sceptical about "most fundamental particle"!)
It's against the spirit of empiricism to go around believing in your models, in particular because you're less likely to come up with a new model if you do. However, we're all human, and believing things we like the sound of incredibly readily is what we do.
How to invalidate your Bayesian argument
We've touched on how testing the consequent is intrinsically flawed logically (which doesn't make it bad science), so that we're never testing truth, we're testing accuracy, and at some examples where those are at variance. Now let's look at some popular other ways to logically invalidate your conclusions.
The numerical Bayesian conclusion after one application depends heavily on the assumed probabilities for your competing theories. This is why Bayesian statisticians like to refine their model time and time again, incorporating new evidence (whether inconsistent with previous data or not) into the calculation. You can come to radically erroneous conclusions by making radically erroneous assumptions at the start.
Even slight or subtle errors can flip your conclusions. In real life, an erroneous initial assumption that a black box would be transmitting normally wasted lots of time and money finding a plane crash, because whilst later searches had been thorough, the model hadn't recorded that the first pass over nearby areas had just checked for the signal to speed things up. Correcting this altered the probability map substantially, and the wreck was quickly found in one of the new Bayesian hot spots.
Sadly, many people attempting to argue from evidence erroneously make drastic initial assumptions like P(Y)=0. (eg "I refuse to entertain that as a possibility without first seeing direct evidence" sounds reasonable to many.) This is both unnecessary and counterproductive, since it invalidates the very statistical argument they're trying to make and makes their reasoning circular, boiling it down to "no, that's nonsense" as both assumption and conclusion.
Another pitfall is to fail to consider possible overlaps and hence probabilistic interactions between your theories, credit to Chris Lively for this answer pointing that out.
Science is not about truth, it's about accuracy. Empirical reasoning makes great science and concludes science is usually most accurate, but that's circular reasoning.
...which doesn't necessarily mean the conclusion is false! ((A-/->B)-/->not B) Note that empiricism concludes that the current scientific model is likely to be proven demonstrably false at some point in the possibly distant future, and replaced by something more accurate, not necessarily closer to the truth.