We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$ be a hyperplane that passes through the center $\mathbf{0}$ of $S$ and has $\mathbf{h}$ as normal vector. Let $E_x$ be the event that $H$ separates $\mathbf{x}$ from the other two points $\mathbf{y}$ and $\mathbf{z}$.

Questions: What is the probability of $E_x$? How can we extend this result to express the probability of separating $\mathbf{x}$ from other $n-1$ points when there are $n$ points instead of only three?

Note: I conjecture that the answer to the first question is $\frac12\left(\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)+\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)-\arccos\left(\mathbf{y}^{\top}\mathbf{z}\right)\right)$.

We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$ be a hyperplane that passes through the center $\mathbf{0}$ of $S$ and has $\mathbf{h}$ as normal vector. Let $E_x$ be the event that $H$ separates $\mathbf{x}$ from the other two points $\mathbf{y}$ and $\mathbf{z}$.

Questions: What is the probability of $E_x$? How can we extend this result to express the probability of separating $\mathbf{x}$ from other $n-1$ points when there are $n$ points instead of only three?

Note: I conjecture that the answer to the first question is $\frac12\left(\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)+\arccos\left(\mathbf{x}^{\top}\mathbf{z}\right)-\arccos\left(\mathbf{y}^{\top}\mathbf{z}\right)\right)$.

We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$ be a hyperplane that passes through the center $\mathbf{0}$ of $S$ and has $\mathbf{h}$ as normal vector. Let $E_x$ be the event that $H$ separates $\mathbf{x}$ from the other two points $\mathbf{y}$ and $\mathbf{z}$.

Questions: What is the probability of $E_x$? How can we extend this result to express the probability of separating $\mathbf{x}$ from other $n-1$ points when there are $n$ points instead of only three?

Note: I conjecture that the answer to the first question is $\frac12\left(\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)+\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)-\arccos\left(\mathbf{y}^{\top}\mathbf{z}\right)\right)$.

We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$ be a hyperplane that passes through the center $\mathbf{0}$ of $S$ and has $\mathbf{h}$ as normal vector. Let $E_x$ be the event that $H$ separates $\mathbf{x}$ from the other two points $\mathbf{y}$ and $\mathbf{z}$.

Questions: What is the probability of $E_x$? How can we extend this result to express the probability of separating $\mathbf{x}$ from other $n-1$ points when there are $n$ points instead of only three?

Note: I conjecture that the answer to the first question is $\frac12\left(\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)+\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)-\arccos\left(\mathbf{y}^{\top}\mathbf{z}\right)\right)$.