Probability of line intersecting the convex set. I would like to prove this theorem:
Let $A,B \subseteq \mathbb{R} ^3$ be convex, limited sets. $B \subseteq A$. I have a "random line", which intersects A. Probability, that this line also intersects B, should be $surfaceArea(B)/surfaceArea(A)$ .
I would like to "prove" that statement. I will probably need to "measure" number of lines or something like that. Do you have any ideas?
 A: I believe this is a theorem of Crofton, the original paper is here (under paywall):
http://www.jstor.org/stable/10.2307/108911
A generalization can be found here:
http://www2.imperial.ac.uk/~rcoleman/secants.pdf
A: What you conjectured makes sense in a natural context and is well known.
As pointed out by Alex R. in comments, this can make sense only if you define the distribution of the line.
One classical possibility is to consider the Haar measure $\mu$ on the space of lines:
Translations and rotations act on the affine lines of $\mathbb{R}^3$ and up to normalization there exists a unique measure on the space of lines which invariant under these actions.
For a convex body (=closed, convex and bounded) $K$, we consider the set of lines intersecting $K$, namely
$$\mathcal{D}_K=\{D \text{ line in }\mathbb{R}^3\mid D\cap K\neq\emptyset\}.$$
The (generalized) Crofton formula (e.g. see theorem 5.1.1. of Stochastic and Integral Geometry) tells us that the Haar measure $\mu(\mathcal{D}_K)$ is proportional to the surface area of $K$. If I remember correctly, this comes from the observation that almost all lines intersecting $K$, intersect twice its boundary.
In this setting it is clear that the probability is 
$$\mathbb{P}(D\cap B\neq\emptyset\mid D\cap A\neq\emptyset)=\frac{\text{SurfaceArea}(B)}{\text{SurfaceArea}(A)}.$$
Note 1: The dimension here is not important and this hold in any dimension.
Note 2: The reference I gave for the (generalized) Crofton formula gives actually a much general property. It relates, the measure of the set of $k$-affine subspaces intersecting a $d$-dimensional convex body $K$, with the $(d-k)$ intrinsic volume of $K$. Well the property is even more general but this goes fare from the original question.
