Exercise 25 Page 935

We can use geometric models to solve certain types of probability problems. In geometric probability, points on a segment or in a region of a plane represent outcomes. The geometric probability of an event is a ratio that involves geometric measures such as length or area. Consider the given diagram.

We are told that a point in the figure is chosen at random, and want to find the probability that the point lies in the shaded region. The probability that the point is in the shaded region is the ratio of the area of the shaded region to the area of the figure. We will find the area of the shaded region and the area of the entire figure one at a time. Then, we will find their ratio.

Area of the Figure

The figure is a circle with a diameter of $5$ units. Since the diameter equals $5$ units, the radius of the circle is $\frac{5}{2}=2.5.$ Let's substitute this value for $r$ in the formula for the area of a circle.