In order to find geometric probability, we will use the circular dartboard shown below.

Suppose that all the points shown on the dartboard above are equally likely to be hit by a dart that you throw.

Notice the followings:

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

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

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

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

Find the following geometric probabilities

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

**2.** You score exactly 3 points

**3.** You score at the most 6 points

**4.** The dart lands inside the red circle

**Solution**

**1.** **P(You score at least 6 points)**

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

P(You score at least 6 points) = area of circle with radius 3r / area of circle with radius 4r

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

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

P(You score at least 6 points) = 9 / 16 = 0.5625 or 56.25%

The probability that you score at least 6 points is 56.25%

**2.** **P(You score exactly 3 points)**

When you score 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 = π(4r)^{2} - π(3r)^{2} = 16πr^{2} - 9πr^{2} = 7πr^{2}

P(You score exactly 3 points) = A / area of circle with radius 4r

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

P(You score exactly 3 points) = 7πr^{2} / 16πr^{2}

P(You score exactly 3 points) = 7 / 16 = 0.4375 or 43.75%

The probability that you score exactly 3 points is 43.75%

**3.** **P(You score at the most 6 points)**

When you score at the most 6 points, your score is 3 points or 6 points.

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

We need to find P(You score exactly 6 points)

When you score 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} - π(2r)^{2} = 9πr^{2} - 4πr^{2} = 5πr^{2}

P(You score exactly 6 points) = B / area of circle with radius 4r

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

P(You score exactly 6 points) = 5πr^{2} / 16πr^{2}

P(You score exactly 6 points) = 5 / 16 = 0.3125 or 31.25%

The probability that you score exactly 6 points is 31.25%

**4.** **P(The dart lands inside the red circle)**

P(The dart lands inside the red circle) = area of circle with radius r / area of circle with radius 4r

P(The dart lands inside the red circle) = π (r)^{2} / π (4r)^{2}

P(The dart lands inside the red circle) = πr^{2} / 16πr^{2}

P(The dart lands inside the red circle) = 1 / 16 = 0.0625 or 6.25%

The probability that the dart lands inside the red circle is 6.25%