Basic question about definition of random polytopes

I am trying to read about random polytopes, the convex hull of n random points $$x_1,\ldots, x_n$$ chosen independently inside a convex body $$K$$ with respect to uniformly distribution. My questions are:

1. Does "uniformly distribution" mean that if we choose $$x_1,...,x_n$$ in a convex body $$K$$, then for any $$Q\subset K$$, we have $$\text{Prob}(x_i\in Q)=\dfrac{V(Q)}{V(K)}$$? Here $$V$$ denotes volume and $$\text{Prob}$$ means probability.

2. With this distribution, why "with probability one, the random polytope is simplicial"?

3. Is there any other distribution considered to create random polytopes?

I would be grateful for any hint on references or literature.

For 1, any (Lebesgue) measurable $$Q$$ will do.
In 2, note that the probability that $$n+2$$ points lie in an affine $$n$$-subspace is zero.