Find geometric probability

 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.

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%