Chapter 13.3, Problem 27PPS

To determine

To prove:

The probability that randomly chosen the point in the circle will lies in shaded region is equal to $\frac{x}{360}$

Explanation of Solution

Given information:

The point in the sector is equal to $\frac{x}{360}$ .

Proof:

The circle has radius $r$ and it contains the sector described by $x$ degrees. The probability that randomly chosen the points lies in the segment is equal to the area of sector is $\frac{x}{360}\pi r^2$ divided by the total area of circle is $\pi r^2$ .

  $\begin{array}{l}P\left(\text{Point}\text{lies}\text{in}\text{sector}\right)=\frac{\frac{x}{360}\pi r^2}{\pi r^2}\\ P\left(\text{Point}\text{lies}\text{in}\text{sector}\right)=\frac{x}{360}\end{array}$

Hence, the probability that randomly chosen the point will lies in shaded region is equal to $\frac{x}{360}$ .