Let us consider the experiment of choosing at random a chord of the circle in $\mathbb{R}^2$ given parametrically by $(\cos 2\pi t,\sin 2\pi t )$, $0 \leq t < 1$. Let $\Omega$ be the unit square $[0,1)^2$. Let $\mathcal{A}$ denote the borel fied of subsets of $\Omega$ and let $P$ be the uniform distribution on $\Omega$. Let $\Psi$ be the space of chords of the given circle and let $\mathcal{B}$ denote the Borel field of subsets of $\Psi$ with $\Psi$ being regarded as a subspace of the metric space of all compact subsets of $\mathbb{R}^2$ (with the Hausdorff metric).
For $w=(w_1,w_2)\in \Omega$, let $X_1(w)$ denote the line segment having endpoints $(\cos 2\pi w_1,\sin 2\pi w_1)$ and $(\cos 2\pi w_2,\sin 2\pi w_2 )$. Notice that $X_1(w)$ is a chord of the circle of interest and $X_1$ is a random set that is, a set-valued random variable.
And let $Q_1$ denote the distribution of $X_1$.
My question is How can I calculate $Q_1(C)$ where $C$ is the set of chords that intersect both the positive vertical axis and the negative horizontal axis. Could someone help me, please?
Thanks for your time and help. (some hints please!)
There is a lot of probability jargon here! In particular it's unclear what is meant by $Q_1$. At any rate your problem seems to be finding the set $\hat C\subset\Omega$ corresponding to segments intersecting the positive $y$-axis as well as the negative $x$-axis.
Let $e^{i\phi_1}$ and $e^{i\phi_2}$ be two points on the unit circle. Then $C$ occurs iff $$0<\phi_1<{\pi\over2}\qquad\wedge\qquad \pi<\phi_2<{3\pi\over2}\qquad\wedge\qquad \phi_2-\phi_1<\pi\ ,\tag{1}$$ or $(1)$ with $\phi_1$ and $\phi_2$ interchanged. The conditions $(1)$ together with their counterparts obtained by $\phi_1\leftrightarrow \phi_2$ define two small triangles in $\Omega$ of area ${1\over32}$ each (draw a figure!). It follows that the probability of $C$ comes to $2\cdot{1\over32}={1\over16}$.