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:
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%
Sep 17, 23 09:46 AM
Jun 09, 23 12:04 PM