Exercise 13 Page 902

We can use geometric models to solve certain type 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. P(The point is in the shaded region)= [0.8em] Area of the shaded region/Area of the figureWe 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 Shaded Region

We can see that the shaded region consists of one inner semicircle and one outer ring. Remember that the area of each semicircle is equal. Let's now consider the given diagram!

Notice that the area of the shaded region is equal to half of the area of the larger circle. A=1/2π r^2 Since the diameter of the circle is 10, then its radius is 102= 5. Let's substitute this value into the above formula.

The area of the shaded region is 12.5π. Shaded Area: 12.5π

Area of the Figure

The figure is a circle with a radius equal to 5. To find its area, we need to multiply the square of the radius by π. Area of the Figure: ( 5^2)π=25π

Probability

As previously mentioned, 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. Since we already know both areas, we can find their ratio.

The probability that a point from the figure chosen at random is in the shaded region is 12, which can be also written as 0.5 or 50 %.