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.
Area of the Shaded Region
Notice that the shaded region is a difference between the larger and the smaller square. Let's first focus on the larger square.
Since half of the side length of the square is 4, the side length is 4(2)= 8. Let's substitute this value in the formula for the area of a square.
The area of the larger square is 64. We can now calculate the area of the smaller square. Notice that the shaded region consists of four right isosceles triangles. Let's now focus on one of these triangles!
Since the hypotenuse of the triangle is also a side of the smaller square, we can use the Pythagorean Theorem to calculate the length of that side.
a^2+b^2=c^2
Let's substitute a= 4 and b= 4 in the above formula, and solve for c.
The area of the smaller square is 32. To find the area of the shaded region, which is formed by the difference between the larger and smaller square, we need to subtract A_S from A_L.
The shaded area is equal to 16.
Shaded Area: 16 square units
Area of the Figure
The figure is a square with side length equal to 8. We calculated earlier that its area is equal to 32.
Area of the Figure: 32 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.
