What is the probability of a symmetric coin fall 
The prompt: We have 10 coins and 1 of them is non-symmetric (with probability of head equal $\frac{1}{3}$). We toss a randomly selected coin 6 times, and obtain 3 tails. What is the probability that we tossed a symmetrical coin?

The way I went on about solving the problem was first assuming it to be a win or success if we obtain a Tails, assuming that
NS = Non symmetric coin, S = Symmetric coin, H = heads, T = tails
$$P(S) = \frac{9}{10}; \text{  } P(NS) = \frac{1}{10}; \text{  }
P(T|S) = \frac{1}{2}; \text{  }P(T|NS) = \frac{1}{3}$$
Now Probability of getting a Tail can be obtained using total probability theorem,
$$P(T) = P(T|S)\cdot P(S) + P(T|NS) \cdot P(NS)$$
$$P(T) = \frac{1}{2}\cdot \frac{9}{10} + \frac{1}{3} \cdot \frac{1}{10} 
= \frac{29}{60}$$
Using Bernoulli's trials, where winning/success(p) is defined as getting a tails, given by $P(T) = \frac{29}{60}$ and failure(q) is defined by getting a head, $P(H) = \frac{31}{60}$ and n = number of tosses.
$$P(X = x) = {n \choose x} (p)^n \cdot (q)^{n - x}$$
$$P(X = 3) = {6 \choose 3} (\frac{29}{60})^6 \cdot (\frac{31}{60})^3$$
Assuming this to be an event E, we now have $P(E) = {6 \choose 3} (\frac{29}{60})^6 \cdot (\frac{31}{60})^3$, 
The desired result is $P(S|E)$, I'm not sure how to go on about finding that, any hint would be much appreciated.

Borrowing your notation, the value we want is: $P(S \mid E) = \frac{P(E \mid S)~P(S)}{P(E)}$, which is Bayes' rule.
Note that:
$$P(E \mid S) = {6 \choose 3} \left( \frac{1}{2} \right)^6 = 0.3125$$
$$P(E \mid NS) = {6 \choose 3} \left( \frac{1}{3} \right)^3 \left(\frac{2}{3} \right)^3=0.219$$
Thus, $P(E) = \sum_i P(E \mid C_i)P(C_i)=\frac{1}{10} \cdot 0.3125 + \frac{9}{10} \cdot 0.219 = 0.0566$. Plug these numbers into the quotient above to get the answer.
