More Geometric Probability

Objective: The student will be able to determine the probability of an event involving area.

When playing games at the fair, it is usually required to hit a target with an item (like a dart), in order to win a prize. What is the probability of hitting the target?

A common game is darts. What is the probability of randomly throwing a dart such that it hits within the red area, given that the dart will always land within the boundary of the outer circle?

In order to find the probability, it is necessary to apply the formulas for areas discussed in Lesson 1 and 2.

The probability will be found by finding the area of the region that is considered a success (the red area) divided by the sample space which is the region contained by the outer circle.

In order to solve this problem, you must know the radius of the circle.

If the radius of the circle shaded red is 1 and the radius of the sample space circle is 5, the Probability of landing in the red region is

P(Red)= Area of Red/ Area of Outer Circle

The figures do not always have to be the same.

In this example, the outer region is a rectangle and the target area is a circle. To find the probability of randomly hitting the target area with a dart,

P(blue) = area of circle/area of rectangle

If the radius of the circle is 1 cm, the length of the rectangle is 5cm, and the width is 2.5 cm.

P(blue)

In order to find the probability of a multiple of regions, it is necessary to add the regions then divide by the sample space.

In this example, if we wanted to find the probability of landing in the yellow region, we could do the following: find the area of each circle, add the areas together and then divide by the area of the square that contains the circles.

How can we do this without knowing the measures of the sides or the radius?

The circles are tangent to each other and to the square. The radius is perpendicular to the tangent line at the point of tangency so the height of the polygon is equal to 2 diameters as is the base.

The diameter is equal to 2(radius). Let r = the radius of the circle. The area of the circle is r(r)(pi). There are four circles so the area of the shaded region is 4(r)(r)(pi).

To find the area of the square, write it in terms of the radius of the circles. The base is 4r and the height is 4r. The area of the square is 16(r)(r).

The probability of the shaded region

Regardless of the radius of the circle, this would be the probability of landing in the shaded region.

This can be verified by changing the length of the radius on the GSP sketch and calculating the probability ratio.

When a skydiver jumps from an airplane, she usually has a landing area already in mind before the jump. What if she wanted to land in the target below? What is the probability that she will hit the target (an equilateral triangle with sides of 2 m), asuuming that she is certain to land within the 10m x 10m square.

The area of the success is the area of the triangle.

The sample space is the area of the square.

The Probability of Landing on the target is

This would be a target left to an expert because the probability is less than 2%.

Practice Problems

Find the probability of the blue shaded regions.

1.

2.

3. If you throw a dart randomly at the target shown, what is the probability that you will hit the shaded area?

4. Mrs. Hollandsworth finds the above targets boring so she designs her own dart board on a 2 feet square board as shown below. If you throw a dart randomly at this strange target, what would be the probability that you land in the interior purple region (notice this region is not a polygon)?

If the dart was certain to land in the purple or yellow region, how would the above probability change? In other words, what is the probability of the dart randomly landing in the purple region given it must land within the yellow region?