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