The quantum key establishment protocol
The protocol is formally known as BB84 protocol.

Your "assumption" are conditional probabilities, and they should be calculated, not "assumed". For example, let Alice sends bit $0$; what is the probability that Bob receives bit $0$? Bob has 2 equally likely choices: either his basis coincides with the Alice's basis or not; in the 1st case Bob receives $0$ bit with probability $1$, in the 2nd case Bob receives $0$ bit with probability $1/2$. So the conditional probability that Bob receives $0$ bit given Alice sent $0$ bit is
$$ P_{X,Y}(0,0|X=0) = 1/2\cdot 1 + 1/2\cdot 1/2 = 3/4$$

Since Alice sends $0$ and $1$ bits equally likely,
$$ P_{X,Y}(0,0) = 1/2\cdot 3/4 = 3/8$$
and this I believe is one of the 4 probabilities that should be inserted into the Joint Entropy Formula