What's the difference between a random and unknown variable? A friend and I were discussing the difference between a random and unknown variable in probability and statistics. The example I used was the shell game, where a marble is placed under one of three shells and the shells are swapped so that we don't know where the marble is. 
The question was what is the probability the marble is underneath the leftmost shell?
I thought the probability was either 1 or 0 but unknown. It can't be 1/3 because then there would be a 1/27 chance every shell has a marble underneath it, or a 8/27 chance there is no marble. Obviously, this doesn't make since in this context.
A different question would be suppose you choose a shell at random, what's the probability that it has the marble?
I thought this would be 1/3 because average probability across all shells is (0+0+1)/3 = 1/3
Is my understanding correct? And, more importantly, why is it important to understand the difference between a random and an unknown variable in probability and statistics?

 A: The frequentist interpretation of this situation would be: if you and your friend did this many times --- you turn your back and she swaps the shells randomly and you pick, then she turns her back and you swap the shells randomly and she picks, no one breaks the rules, and you tally the outcomes over time --- then 
$$
\lim_{N \rightarrow \infty} \dfrac{1}{N} \sum_{i=1}^N \mathbb{I} \{ \text{The shell with the marble was picked in trial $i$}\} = \dfrac{1}{3}.
$$
Because we keep using the word "random", it imposes some order on the world.
But if you were only doing this experiment once and there were no guarantee of "random", impartial assignment of the marble, there would be no reason to assign a particular probability mass function to the experiment. Instead, it would make sense for you to have many possible probability mass functions, that are all consistent with the experiment. Perhaps you know your friend hurt her right arm and is right handed, so is more likely to put the marble under the shell closest to her right hand rather than stretch out to the leftmost shell.  This is usually called a multiple priors model, or sometimes "ambiguity", or something the difference between risk and uncertainty.  
The best example is the Ellsberg paradox. Daniel Ellsberg showed that when you give people objective odds over something, they behave differently from when you refuse to do so. In his experiments, he would give people bets like, "In gamble A, the urn has 10 red balls and 10 blue balls; if you draw a red ball I will give you ten dollars. In gamble B, the urn has 20 red balls and 10 blue balls; if you draw a red ball I will give you 3 dollars.  Which gamble would you prefer?" In that formulation, people would behave in ways consistent with standard models of decision under uncertainty, like frequentists. However, when he gave people bets like, "In gamble A, the urn has 10 red balls and 10 blue balls; if you draw a red ball I will give you ten dollars. In gamble B, the urn has twenty balls, but I will not tell you the color; if you draw a red ball I will give you 7 dollars.  Which gamble would you prefer?" the results changed very much, as people hated the ambiguity of the odds on Gamble B.  It seemed different from the objective risk of the first set of gambles. 
So the answer might be: If you are a frequentist, the marble is under every shell with probability 1/3, but if you have not taken a college-level statistics class, any prior about which shell contains the ball is reasonable.
A: No.  You cannot assign a positive probability to the outcomes where there is one marble under each shell, or no marbles under any shell, because those are impossible.  The probability is $1/3$ under an assumption that the shuffling is at random.  This is the same as the second case, except the randomness in the second case comes from your choice of shell.  But in both cases, there is an assumption of randomness, which is why a probability can be assigned.
An example in which we draw a distinction between a random variable, and an unknown variable, is the following.  Suppose I have a bag containing a large number of coins of various denominations.  I want to know the total weight of the coins to the nearest gram.  Of course, this is not hard to determine; if we have a precise enough scale for the amount of weight, we can simply weigh the coins.  We can even take this result and divide by the number of coins in the bag to get an average weight of the coins in that bag.
But now, say we want to know the total weight of all coins in the world, or maybe the average weight of all coins in the world.  From a mathematical standpoint, there is no difference in the computation; it is simply a difference in the choice of population (the bag, versus the whole world).  If you could gather all of the coinage and weigh it all, and do it at a single point in time, you could precisely ascertain this value.  There is no randomness to it, and its value will not change unless a new coin is made, or an existing coin is defaced or destroyed.  But for many reasons, it is impractical to actually perform this measurement, unlike the earlier example with only a bag of coins.  So this quantity is an unknown variable.  In statistics, we call fixed but unknown quantities parameters, and we are often interested in the estimation of such parameters.
To this end, our intuition suggests that if we sample a subset of coins from the world's population of coins, and measure those, we might get a reasonably close approximation of the true average weight of the world's coins.  But here is where some additional consideration is required, for not all of the world's coins are made in the same way; some countries make heavier coins than others, for instance.  So when we take such a sample, we must ensure that it is somehow representative of the whole, in order for our estimation to be successful.  Another consideration is that we must take enough coins in our sample so that we are not led astray by a single unusually light or heavy coin.
The main remedy in both cases is what is called randomized sampling.  That is to say, we do not attempt to systematically or selectively take certain coins for weighing than others.  By being naive to any particular choice, we hope to achieve a representative sample.  This introduction of randomness occurs in the sampling process itself, and is not an intrinsic property of the coins, each of which has a fixed weight.  So this is why if, for instance, you take a random sample and use it to estimate the average weight of the world's coins, your estimate may be different than if I performed the same experiment, because from one experiment to another, the actual coins we weighed are not the same.
What this tells us is that the observed weight from experiment to experiment is random because the sampling is random, and that this can be regarded as a random variable.  Even though it aims to approximate the unknown variable (the parameter), each time we do the experiment, the result changes.  We can then assign probabilities to outcomes such as the average coin weight of the sample being less than 5 grams, or the average coin weight of the sample being between 3.5 and 3.7 grams.  But it is not a statement about the true weight we are making here; because that isn't random.  We are talking about the sampled weight which is random.

In summary, it is important to understand the source of randomness when considering notions of probability, random variables, parameters, estimation, experiments, and inference.  We do not ascribe randomness to quantities that are fixed, even though though they may be unknown or unknowable.
