Question

Hello I need help Finding the probability

Answer 1

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
`\frac\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}}=\frac{25\pi}4-\frac{25}2`

This means the area of one petal is
`\frac{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)=(50\pi-100)/(100)=\frac\pi2-1`

`\mathbb P(X=G)=(200-50\pi)/(100)=2-\frac\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`