different interpretations?

Suppose you have a 2-headed coin, a 2-tailed coin, and a fair coin. You choose a coin randomly. What is the probability that you get heads? What is $\displaystyle P \left(\text{2-headed coin}| \text{heads} \right) $

This is similar to the Monty-Hall problem right? Namely, you have $\displaystyle P(H) = \begin{cases} 1 \ \ \text{if 2-headed coin} \\ 0 \ \ \text{if 2-tailed coin} \\ \frac{1}{2} \ \ \text{if fair coin} \end{cases} $.

The answers are $\displaystyle \frac{2}{3} $ and $\displaystyle \frac{1}{3} $ respectively?

But is this only in the classical interpretation? Can we use another interpretation and get another answer?