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 Figure
The figure is an equilateral triangle. Let's consider the given diagram once again!
We know that the side length of the triangle is equal to 14, that all angles in the triangle are congruent, and that each of them measures 60^(∘). Now, notice that by using the sine ratio we can find the height h of the triangle.
We now know that the side length of the small triangle is equal to 4, that all angles in the triangle are congruent and that each of them measures 60^(∘). Notice that by using the sine ratio we can find the height h_a of the small triangle.
The area of one small triangle is 4sqrt(3). To find the area of the shaded region, which is formed by the difference between the large triangle and three small triangles, we need to multiply A_S by 3 and subtract it from A_L.
The shaded area is equal to 37sqrt(3).
Shaded Area: 37sqrt(3)
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.
