Each petal of the region \(R\) is the intersection of two circles, both of diameter 10. Each petal in turn is twice the area of a circular segment bounded by a chord of length \(5\sqrt2\), which implies the segment is subtended by an angle of \(\dfrac\pi2\). This means the area of the segment is
\(\text{area}_{\text{segment}}=\text{area}_{\text{sector}}-\text{area}_{\text{triangle}}\)
\(\text{area}_{\text{segment}}=\dfrac{25\pi}4-\dfrac{25}2\)
This means the area of one petal is \(\dfrac{25\pi}2-25\), and the area of \(R\) is four times this, or \(50\pi-100\).
Meanwhile, the area of \(G\) is simply the area of the square minus the area of \(R\), or \(10^2-(50\pi-100)=200-50\pi\).
So
\(\mathbb P(X=R)=\dfrac{50\pi-100}{100}=\dfrac\pi2-1\)
\(\mathbb P(X=G)=\dfrac{200-50\pi}{100}=2-\dfrac\pi2\)
\(\mathbb P((X=R)\land(X=G))=0\) (provided these regions are indeed disjoint; it's hard to tell from the picture)
\(\mathbb P((X=R)\lor(X=G))=\mathbb P(X=R)+\mathbb P(X=G)=1\)
