We can use geometric models to solve certain types of probability problems. In geometric probability, points in a region of a plane represent outcomes. The geometric probability of an event is a ratio that involves geometric measures such as area. Let's first draw a diagram, knowing that a circle is contained inside a square.
We are told that a point in the figure is chosen at random, and want to find the probability that the point in the interior of the square lies in the interior of the circle, which is also 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 know that a circle with radius 3 is contained inside a square with a side length equal to 9. Let's first focus on the circle!
Since the radius of the circle is 3, we can substitute this value into the formula for the area of a circle.
The area of the shaded region is 9Ï€.
Shaded Area: 9Ï€
Area of the Figure
The figure is a square with a side length equal to 9. To find its area, we need to find the square of the side length.
Area of the Figure: 9^2=81
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.
