We can use geometric models to solve certain types of probability exercises. 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.
Area of the Figure
The figure is a rectangle. We also know that the radius of each circle is 3 units. Therefore its diameter is 3(2)= 6 units. Notice that there are exactly three circles along the length and two circles along the width of the rectangle. This means that the length of the rectangle is 6( 3)= 18 units and its width is 6( 2)= 12 units.
Since the length l of the rectangle is 18 and its width w is 12, we can substitute these values in the formula for the area of a rectangle.
The area of the rectangle is 216 square units.
Area of the Figure: 216 square units
Area of the Shaded Region
We know that the area of the shaded region is the difference between the area of the rectangle and the area of the circles. Let's focus on one of these circles. Since each circle has radius of 3 units, we can substitute this value in the formula for the area of a circle.
The area of one circle is 9Ï€.
To find the area of the shaded region, which is formed by the difference between the rectangle and sixcongruent circles, we need to multiply A_c by 6 and subtract it from A_r.
The shaded area is equal to about 46.35 square units.
Shaded Area: 46.35 square units
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.
