To find the geometric probability, we will use the circular dartboard shown below.

For example, suppose that all the points shown on the top of the dartboard are equally likely to be hit by a dart you have thrown.

Please pay attention to the following:

- If your dart hits the dartboard outside the blue circle but inside the black circle, your score is 3 points.

- If your dart hits the dartboard outside the green circle but inside the blue circle, your score is 6 points.

- If your dart hits the dartboard outside the red circle but inside the green circle, your score is 12 points.

- If your dart hits the dartboard inside the red circle, your score is 24 points.

Find the following geometric probabilities

**1.** You get at least 6 points

**2.** You get exactly 3 points

**3.** You get a maximum of 6 points

**4th** The arrow lands in the red circle

**solution**

**1.****P (you get at least 6 points)**

If you get at least 6 points, your score will be 6 points, 12 points, or 24 points. The radius in this case is 3r.

P (you get at least 6 points) = circular area with radius 3r / circular area with radius 4r

P (you get at least 6 points) = π (3r)^{2} / (4r)^{2}

P (you get at least 6 points) = 9πr^{2} / 16πr^{2}

P (you achieve at least 6 points) = 9/16 = 0.5625 or 56.25%

The probability that you will get at least 6 points is 56.25%

**2.****P (you get exactly 3 points)**

If you get exactly 3 points, you need to find the area outside the blue circle but inside the black circle.

Let A be the area outside the blue circle but inside the black circle.

A = (4r)^{2} – (3r)^{2} = 16πr^{2} – 9πr^{2} = 7πr^{2}

P (you get exactly 3 points) = A / circular area with radius 4r

P (you get exactly 3 points) = 7π (r)^{2} / (4r)^{2}

P (you get exactly 3 points) = 7πr^{2} / 16πr^{2}

P (you get exactly 3 points) = 7/16 = 0.4375 or 43.75%

The probability that you will get exactly 3 points is 43.75%

**3.****P (you get a maximum of 6 points)**

If you get 6 or less, your score will be 3 or 6 points.

We already have an answer for getting 3 points. It is 43.75%

We need to find P (you get exactly 6 points)

If you get exactly 6 points, you need to find the area outside the green circle but inside the blue circle.

Let B be the area outside the green circle but inside the blue circle.

B = π (3r)^{2} – (2r)^{2} = 9πr^{2} – 4πr^{2} = 5πr^{2}

P (you get exactly 6 points) = B / circular area with radius 4r

P (you get exactly 6 points) = 5π (r)^{2} / (4r)^{2}

P (you get exactly 6 points) = 5πr^{2} / 16πr^{2}

P (you get exactly 6 points) = 5/16 = 0.3125 or 31.25%

The probability that you will get exactly 6 points is 31.25%

**4th****P (the arrow lands in the red circle)**

P (the arrow lands in the red circle) = circular area with radius r / circular area with radius 4r

P (the arrow lands in the red circle) = π (r)^{2} / (4r)^{2}

P (the arrow lands in the red circle) = πr^{2} / 16πr^{2}

P (the arrow lands in the red circle) = 1/16 = 0.0625 or 6.25%

The probability that the arrow will land inside the red circle is 6.25%