Interpreting result of experiment with no prior hypothesis I've just signed up to this site to ask this question, so apologies ahead if I'm not obeying your standards. Although I will ask some concrete questions, I have a feeling this might lead to open ended discussion so if this is not what you want I'm okay with moderator erasing this, but I'd appreciate directions on where to post this.
Also, I intentionally wrote this in "easy language" to make my thoughts more understandable. I am familiar with concepts and terminology of statistics so don't hesitate to use them in answer should you find it necessary. 
 Problem 
Suppose I approach you with a coin. I flip it in front of you $10$ times, and it falls on heads each time. What will you think about that coin? (For the sake of convinience, assume $10$ was enough to make you believe that the coin is unfair, if not, replace $10$ with sufficiently large number that will make you happy believing the coin is unfair).
Okay, having established that you think that the coin is unfair, I perform following..
I take out $1000$ other coins from my bag and flip each and every one of them in front of you $10$ times. The distribution they form turns out to be as you would expect of them, and, say that coin number $337$ happened to fall $10$ times on heads.
 -- questions:  What will you think about coin $337$?
What will you now think about first coin I approached you with?
 Motivation 
More or less obviously, this question is inspired by various researches performed nowadays.
Suppose you are a doctor trying to discover new cure. You do some testing, trial and error like experiments and come up with a cure that cured $99$% of people. But in the meantime you didn't publish any of the experiments when you failed on $50$% of people, so how do I know that this remarkable $99$% success rate is anything more then mere "random luck"?
Of course it's a bit more complicated when it comes to cures because there is no universal distribution that we expect cures to come from as opposed to coins when we expect that probability of heads is roughly $50$%. Another thing is that when doctor obtains $99$% success rate, they will probably redo the testing couple more times to make "sure" this wasn't a mere random luck.
So, don't get me wrong, I don't have doubts about our medicine, I'm just interested in what should a mathematician think about the coin problem I stated.
 Some remarks 
Firstly, I'd like to emphasise that I understand that when performing a statistical experiment we need to have hypothesis and some sort of formal setting, which I didn't provide here, but still I'm seeking for a formal answer. This might seem a bit unfair but I think we encounter experiments as this one all the time, and as mathematicians, should have an opinion about them.
I have several thoughts on this subject so here they go:


*

*Maybe it is generally 'false' to state "suppose this and that happened, what should I think about that", if this and that didn't actually happen. (Note that I didn't actually toss any coin..) I doubt this argument is good, but I stated it anyways in case someone gets inspired by it.

*Perhaps we really cannot conclude anything without having a hypothesis in the first place? This also seems dull - people evolved a lot based on intuitive interpretation of experiments throughout history. Besides, if someone tossed you a coin that fell on heads $10$ times in a row... I'm sure you'd be suspicious about that.
So, given that problem is well stated and that I can conclude something (i.e. arguments 1. and 2. are false), what are our options?
 A)  Thinking that coin $337$ is biased and first coin isn't biased. - this seems a bit stupid.
 B)  Thinking that coin $337$ is biased and first coin is also biased. - possible, but seems unlikely to me. Why would coin number $337$ be biased? Surely one of those $1000$ will land on heads $10$ times.
 C)  Thinking that coin $337$ isn't biased and first coin is biased. - possible, but also seems a bit unlikely to me. Why would you treat first coin any differently then coin $337$?
 D)  Which leaves me thinking that: neither of coins is biased.
Now, how do I go about backing up my opinion? I think this is the way I should be thinking - initially, based on my prior experiences from "world", I have some distribution curve of $p$ for first coin ($p = $ probability that coin flips on head). In my case it would look roughly as a discrete random variable with, say $p = 0.5$ having probability $98$%, and $p = 0$ and $p = 1$ having probability $1$%.
After initial $10$ tosses, my distribution curve for general coins is going to change a bit and I will generate new distribution curve for this coin - it will look something like $p = 1$ having probability $99$% or so.
Now, after another $1000 \times 10$ tosses, my distribution curve for general coin will also change a bit and end up more or less the way it was in the beginning. But, my distribution curve for the first coin is going to change drastically, looking more or less like the one for general coin - because in my mind that experiment is "very" connected to $1000$ experiments later. 
This reasoning also seem intuitive. If someone tosses a coin $10$ times on heads that's cool, but if they do something "ordinary" with next $1000$ coins, then they lose credibility for first coin as well. Similar argument holds for doctors. If you succeed to cure $99$% of people, but just in $1000$th attempt, maybe you are just a bad doctor and the fact you were doing first $999$ experiments tells you that you have bad intuition on what a good cure should be.
Now, this reasoning seems perfectly fine to me from intuitive point of view, but it scares me when it comes to formalising it.
So, suppose reasoning is valid, and after first $10$ tosses I didn't do any other tosses but rather you observed $1000$ things that you think should come from roughly binomial distribution and they did (traffic light was green, minute needle on clock was even, ...) Should you now think that coin is fair after all, just because some other experiments you observed behaved as expected? Surely no.
So if my reasoning is valid, one should define "connectedness" between experiments and this seems very far from being doable in formal mathematical means. It seems highly intuitive that reasoning is valid, but I have no idea how to model it. 
To sum up  -- questions  once more. Is the problem I posed valid and you can conclude something? (i.e., arguments 1. and 2. are false, as well as any other similar arguments) If the problem is well posed, what should you formally, mathematically, think about coin $337$ and what about first coin?
I've read some papers which I can't cite now since I lost track of them (but surely there are quiet a few) concerning the problem with doing research and not publishing unsuccessful results, but none with answer to my question. So any thoughts or references are welcome!
 A: You raise some important issues. But there is too much here to try to respond cogently to all of it in a reasonable
amount of time and space. So I'll pick away at what I see as the main related
issues for research using statistical analysis.
Need for underlying probability structure. First, in the context of medical experiments, you ignore the crucial issue
of a 'protocol' requiring a well-defined population, a random sample from
that population, random assignment of subjects to Treatment and Control
groups, and a specified way to measure outcomes. Randomization lays the groundwork for using probability in statistical inference.
We can test the null hypothesis that $H_0: \mu_T = \mu_C$ against
$H_a: \mu_T \ne \mu_C.$ Here $\mu_T$ and $\mu_C$ are means of
hypothetical populations of subjects treated and not treated, using methods of measurements that are meaningful for outcomes. Suppose that measurements on the
subjects in the two groups give estimates
$\hat \mu_T$ and $\hat \mu_C$ (usually sample means $\bar X_T$
and $\bar X_C$) of the respective population means that cast doubt
on $H_0$ according to some criterion based on the population distributions.
Then we would reject $H_0$.
Conclusion about data not hypotheses. It is important to understand that 'rejection' is really a statement
about the data. The data are very unlikely to have arisen from a situation
in which $H_0$ is true. So we can either believe that a very unlikely
event has occurred OR to suspect that $H_0$ is wrong, and hence that the
treatment had some effect. 
In this framework, any statistical inference about the fairness of
a single coin that shows 10 Heads in a row is on shaky ground. We don't
know the population from which the coin was chosen. It is up to my
personal opinion about the coin. Maybe you work for a company that manufactures
two-headed coins for magicians. Personally, I might say I wouldn't want
to play a gambling game with you based on use of that coin. Or maybe not
with any coin you produce as a substitute.
As for coin #377 out of a population of a thousand coins. If I believe
that is a population of fair coins, I wouldn't be astonished to see
at least one coin with all heads or all tails. (Any one coin has probability
$2/2^{10} = 0.001953125$ of doing that, and you've just demonstrated 1000 coins.) But not knowing anything about your population, I'm still not
willing to do inference. There is no known 'protocol', no well-defined population. Coin #377 might just be your two-headed coin again.
Statistics is inductive, not deductive. Most endeavors in the mathematical sciences are fundamentally deductive.
Start with axioms and see what can be deduced from them. Statistics
is a mathematical science in that it makes heavy use of mathematics, but
much of statistics is fundamentally inductive. We are never going to
know for sure whether any $H_0$ is "true." We can only make statements
about data and try to do informed speculation.
The "File-Drawer" Phenomenon. Because of something called the file drawer phenomenon, informed speculation
can be very difficult in practice. Suppose a medical group does a study
of a new drug following a strict protocol and excellent experimental design,
and rejects "$H_0:$ Drug has no effect" at the 5% level. This may seem persuasive evidence
that the drug has some effect. However, what if we are told that the drug manufacturer paid
50 medical groups to do the same trial on the same drug, and that the reported
"Rejection" is the only one out of 50? (The other 49 are hidden away in a
file drawer.) 
Recently a large-scale retrospective study of many key findings in psychology
found that disappointingly many of published results could not be replicated.
Perhaps some studies were based on poor protocols or experimental design;
perhaps some data were fudged; and perhaps some where the $N$th try to get
significant results while the first $N-1$ remain hidden away. 
Dealing with publication bias. Usually, only 'significant' results get published. Some feel that the standard for publication ought to be based on the 
cogency of the protocol and the design, with 'acceptance' of the paper
before data are collected. Then the paper is published along with whatever
results happened to occur. At least then others might know whether the
study represents a fruitful avenue for future invenstigation.
