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 outside the shaded region. The probability that the point is outside the shaded region is the ratio of the area of the region that is not shaded to the area of the figure.
P(The point is outside the shaded region)= [0.8em]
Area of the region that is not shaded/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 can calculate the area of the region that is not shaded. Finally, we will find their ratio.
Area of the Shaded Region
The shaded region is a circle with diameter 18 centimeters. Therefore, its radius is 182=9 centimeters. Let's substitute this value in the formula for the area of a circle.
Therefore, the shaded area is 81π square centimeters.
Area of the Figure
The figure is a regular hexagon with side lengths of 10.4 centimeters and an apothem of 9 centimeters.
To find the area of this hexagon, we first need to calculate its perimeter. Since this is a regular hexagon, the perimeter is six times its side length.
P=6* 10.4 ⇔ P= 62.4cm
Let's substitute these values in the formula for the area of a regular polygon.
The area of the region that is not shaded is about 26.33 square centimeters.
Probability
As previously mentioned, the probability that the point is outside the shaded region is the ratio of the area of the region that is not shadedto the area of the figure. Since we already know both areas, we can find their ratio.
P=Area of the region that is not shaded/Area of the figure
