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
We know that the shaded region is a difference between a larger regular hexagon and four smaller regular hexagons. Let's first focus on the formula for the area of a hexagon!
Let's draw an apothem a of a hexagon. Since the central angle of a hexagon is 60^(∘) and the apothem bisects this angle along with the corresponding side, the bisected angle is 60^(∘)÷2= 30^(∘).
Let's write and solve an equation for a using the tangent ratio.
The apothem of the hexagon is equal to ssqrt(3)2. Now, we can recall the formula for the area of a regular polygon.
A=1/2ans
In this formula a is the apothem, n is the number of sides, and s is the side length. Since a hexagon is a regular polygon, we can substitute ssqrt(3)2 for a and 6 for n in the formula.
We now know that the formula for the area of a hexagon is 3s^2sqrt(3)2.
Area of the Large and Small Hexagons
Finally, let's focus on the larger hexagon.
We can see on the given diagram that the side of the larger hexagon is equal to 8, which we can now substitute for s in the formula for the area of a hexagon.
The area of the larger hexagon is 96sqrt(3). Let's now focus on one of the smaller hexagons.
We can see on the given diagram that the side of the smaller hexagon is equal 3, which we can now substitute for s in the formula for the area of a hexagon.
The area of one small hexagon is 13.5sqrt(3). To find the area of the shaded region, which is formed by the difference between the large hexagon and four small hexagons, we need to multiply A_S by 4 and subtract it from A_L.
The shaded area is equal to 42sqrt(3).
Shaded Area: 42sqrt(3)
Area of the Figure
The figure is a hexagon with side length equal to 8. We calculated earlier that its area is equal to 96sqrt(3).
Area of the Figure: 96sqrt(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.
