# Why does the point $(0,0,-1)$ on the bloch sphere correspond to the state $|1\rangle$ and not $-|1\rangle$ or $e^{i \phi}|1\rangle$?

In this representation for points on the $$Z$$-axis, $$\phi$$ is not defined. If the point $$(0, 0, 1)$$ is taken since $$\theta$$ is $$0$$ and $$\sin(\theta/2)$$ is zero, it doesn't matter what $$\phi$$ is. The point $$(0, 0, 1)$$ represents state $$|0\rangle$$.

But for point $$(0, 0, -1)$$, $$\theta$$ is $$\pi$$. $$\sin(\pi/2) = 1$$ and $$\cos(\pi/2) = 0$$. So the state is $$e^{i \phi}|1\rangle$$ where $$\phi$$ is not defined.

But I saw that point $$(0, 0, -1)$$ corresponds to state $$|1\rangle$$. Why? There is an equal chance of it being $$-|1\rangle$$ or any other state of type $$c|1\rangle$$ where $$c$$ is a complex number and $$|c| = 1$$.

The $$e^{i\phi}$$ factor you are referring to is called a global phase. You can read more about that here. When you are talking about the state $$-|1\rangle$$, this $$e^{i\pi} = -1$$ factor has no physical relevance. Your probability of outcome doesn't change.
For us, states $$e^{i\phi} | 1\rangle$$ for any value of $$\phi$$ are equilvalent. It is an artefact of the mathematical framework that we use$$^1$$, and there is no physical relevance to this global phase. You cannot measure this quantity physically, nor does it have any effect on your outcomes/measurements.
So every point on the Bloch sphere not only corresponds to a state $$|\psi\rangle$$ but every possible state $$e^{i \phi}|\psi\rangle$$, where $$\phi \in \mathbb{R}$$.
Note: Complex variable $$c$$ with $$|c| = 1$$ that you are talking about, is this $$e^{i\phi}$$ factor.
1: We generally say states are vectors in Hilbert space, however, we actually work with rays in Projective Hilbert space. One state corresponds to a ray where points on this ray are exactly all $$e^{i \phi}|\psi\rangle$$ states for different values of $$\phi$$. You can read more about these here.