expected value: "Consider 12 independent rolls of a 6-sided die...."

I have the following question:
Consider 12 independent rolls of a 6-sided die. Let X be the number of 1's and let Y be the number of Ts obtained. Compute E[ X], E[ Y], Var(X), Var(Y), E[X Y], Var(X Y), Coy (X, Y), and p(X, Y). (Hint: You may want to compute the in the order given.)

I answered the following :

E[ X] = np = 12 * 1/6 = 2
E[ Y] = 2 (same as above)
Var(X) = np(1-p) = 5/3
Var(Y) =5/3 (same as above)
E[X + Y] = E(X) + E[Y] = 4
Var(X + Y) = Var(X) + Var(Y) = 10/3 (independent variables)
Cov (X, Y) = E(XY) - E(X+Y) = ? - 4
p(X, Y) No idea
are my answers correct?

Okay, so I'm assuming that the "number of Ts obtained" is actually a typo and you meant "number of 5's obtained." If that's not right, please reply with any necessary corrections. I'll assume it is for the remainder of my post. I agree with all of your answers up through Var(X+Y). Past that, your answers are not correct. For Cov(X,Y), you might find this theorem sheet from Osaka University helpful. In particular, note Theorems 3 and 4:

Theorem [3]: Cov(X, Y) = E(XY) − E(X)E(Y)

Click to expand...

and

Theorem [4]: Cov(X, Y) = 0, when X is independent of Y

Click to expand...

What does this information suggest the Cov(X,Y) for your problem is? Why? Then for p(X,Y), it looks like you're using a slightly different notation, but if my inference is correct and you're meant to find the correlation coefficient between X and Y, you can also use that sheet from Osaka University to help you out here:

Definition: The correlation coefficient between X and Y, denoted by \(\displaystyle p_{xy}\), is defined as:

\(\displaystyle p_{xy}=\dfrac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)} \cdot \sqrt{\text{Var}(Y)}}=\dfrac{\text{Cov}(X,Y)}{ \sigma_x \cdot \sigma_y}\)

Click to expand...

Theorem [6]:\(\displaystyle ρ_{xy} = 0\), when X is independent of Y

Click to expand...

What does this information suggest the p(X,Y) for your problem is? Why?

Sounds right to me. The amount of 1's rolled in no way affects the number of 5's rolled, so the two random variables X and Y must be independent of each other.