How to interpret probability theory in a deterministic world? Let's assume that our world is deterministic. Consequently, I assume, apparent randomness arises from lack of knowledge. Now consider this example. When you flip a coin, chances of getting tails and heads are 50-50. Why? The equations governing the motion of the coin are known, but the initial conditions are 50-50 in favor of tails or heads. Why? Because when you picked the coin from the table, it was on tails or heads 50-50. You can trace back the initial conditions all the way to... the big bang? 
In short: With determinism taken for granted, what determines the initial conditions? Is there a random element present from the beginning of the world necessarily?
 A: The way the coin faces before the flip may be predetermined, but that doesn't mean that you (or anyone else interested in the result of the flip)  knows what it is (you might pick up the coin without looking at it or letting anyone else look at it). Also, every single muscle twitch that goes into setting the coin spinning might be deterministic, but that doesn't mean that you can control them, or that any of the other players can predict them. Same with air resistance, and how the coin bounces as it hits the table.
In the end, while the result may be entirely deterministic, we as humans do not have enough information and / or calculation power to make a pretiction of the result of the coin flip in any reasonable time. Thus, while it's not truly random, it behaves randomly for any intents and purposes. Most importantly, no one is capable of using what they know to gain any kind of advantage in a coin-flipping game over someone who is purely guessing.
A: The key idea is that our random variable is nothing more than a model for the physical process of the coin. We acknowledge in our construction of the model that it's not perfect. For instance, the printing of an actual coin may give it a raised surface, which may cause a bias in the outcome when it makes contact with your thumb upon being flipped. This bias is incalculable and practically insignificant, so we ignore it in practice. We can study a mathematical model, i.e. $X(\omega)$ for $\omega \in \{a, b\}$, a measure space with $\mu(a) = \mu(b) = 0.5$. That model can only be imperfectly applied to a real physical process, but it's a darned good approximation.
To dig into your questions more deeply:

When you flip a coin, chances of getting tails and heads are 50-50. Why? 

Because experience tells us this is the case.

The equations governing the motion of the coin are known, but the initial conditions are 50-50 in favor of tails or heads. Why? 

I'd argue that the initial conditions are mostly irrelevant. The model of a coin flip is that regardless of how it faces when you pick it up, the act of spinning it will cause it to land on equal head with equal-ish probability.

Because when you picked the coin from the table, it was on tails or heads 50-50. 

As mentioned, I think this assumption is not important, and may in fact be verifiably false.
I am a mathematician, so I study mathematical processes. If someone asks me to say something about the real world, I'll always preface it with, "assume that [math-y assumption]," and I suppose it's up to my audience to decide whether they buy that assumption or not. Such questions are for physicists and philosophers.
