Exercise 27 Page 903

We want to prove that the probability that a randomly chosen point in the circle will lie in the shaded region is equal to x360.

We will use geometric probability. We know that the probability that a randomly chosen point in the circle will lie in the shaded region is equal to the ratio of the circle of the shaded region to the area of the circle. Let's call the length of the radius of our circle r.

Notice that the shaded region is a sector of the circle. The area of a sector of a circle is the product of the area of the circle and the ratio of the measure of the sector to 360. We can see from the diagram that the measure of the shaded sector is x^(∘). Area of the Circle &= π r^2 Area of the Sector &= x/360 π r^2 Let's calculate the probability of a randomly selected point in the circle being in the shaded region.

As we can see, the probability of a randomly selected point in the circle being in the shaded region is indeed equal to x360.