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

Geometric probability with a dart board

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) = 9πr2 / 16πr2

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 = 16πr2 – 9πr2 = 7πr2

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) = 7πr2 / 16πr2

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 = 9πr2 – 4πr2 = 5πr2

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) = 5πr2 / 16πr2

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%

4thP (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) = πr2 / 16πr2

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%



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