Take the Bell state $$\frac{1}{\sqrt{2}}\left(|00\rangle +|11\rangle\right) $$ and measure the first qubit with respect to some axis $\vec{n}_1$ on the Bloch sphere, i.e. measure the observable $$\vec{n}_1\cdot \vec{\sigma}$$ where $\vec{n}_1$ is normalized and $\vec{\sigma}$ is the vector of Pauli matrices.
Now, measure the second qubit with respect to a different axis $\vec{n}_2$.
What is the probability of equal measurement outcomes as a function of the two axes $\vec{n}_1,\vec{n}_2$, or, say, as a function of the angle between the axes?
I have read somewhere that the right answer is $p(\text{same outcome})=\cos^2\left(\frac{\theta}{2}\right)$ where $\theta$ is the angle between the axes.
I am looking for an easy way to obtain this result, or any kind of intuition. I believe I am capable of just carrying out the full-blown calculation myself. In fact, I started but it is very tedious and does not seem to provide me with any intuition about the geometry that's happening here.