Find geometric probability

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

Geometric probability using a dartboard

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) = 9πr2 / 16πr2

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 = 16πr2 - 9πr2 = 7πr2

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) = 7πr2 / 16πr2

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 = 9πr2 - 4πr2 = 5πr2

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) = 5πr2 / 16πr2

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) = πr2 / 16πr2

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%