I think I might be misunderstanding the concept of a simplex and its volume.
Take the 2-dimensional simplex (a triangle) embedded in 3-dimensional space with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. When I calculate its area, I get $\frac{\sqrt{3}}{2}$. However, I've come across information suggesting that the volume (area in this case) of this probability simplex should be $\frac{1}{2}$.
Similarly, the 3-dimensional tetrahedron (a 3-dimensional simplex) embedded in 4-dimensional space with vertices $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, and $(0,0,0,1)$. When I calculate its volume, I obtain $\frac{1}{3}$, but I've read that the volume of this 4-dimensional probability simplex is $\frac{1}{6}$.
Can someone explain the meaning of this discrepancy in the volume?
The area of the 2-simplex is indeed $\sqrt{3}/2$. There's a different 2-simplex, with vertices $(0,0), (1, 0), (0, 1)$ in the plane whose area is $1/2$; the corresponding 3-simplex in 3-space (with vertices at the origin and at all points with exactly one non-zero coordinate, which is 1, i.e. $(1,0,0), (0,1,0), (0,0,1)$) has volume $1/6$. In dimension $n$, that simplex has volume $1/n!$.