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Hello I need help Finding the probability

Hello I need help Finding the probability-example-1
User Eyalyoli
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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
Hello I need help Finding the probability-example-1
User Sahil Deliwala
by
8.6k points

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