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.
P(The point is in the shaded region)= [0.8em]
Area of the shaded region/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.
We found that the area of the figure is 9π square feet.
Area of the Shaded Region
To calculate the area of the shaded region, we need to subtract the area of a circle with diameter 4 feet from the area of the figure. The radius of this circle is 42= 2 feet. Once again, we will use the formula for the area of a circle.
We found that the area of the inner circle is 4π square feet. With this information, we can calculate the shaded area.
Shaded Area: 9π - 4π= 5πft^2
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.
