A simple dartboard has three areas…the main board has a radius of 8 inches, there is a circle with a radius of 6 inches, and the bullseye has a radius of 4 inches. What is the probability of a random dart landing inside the bullseye?
I would say:
probability = area of bullseye/area of main board
Though strictly speaking the question ought to read "What is the probability of a random dart that hits the dartboard landing inside the bullseye?". A truly "random" dart could easily miss - and that would make the probability uncalculatable without extra information on how likely it is to hit the board.
To find the probability of a random dart landing inside the bullseye, we need to compare the area of the bullseye to the total area of the dartboard.
The total area of the dartboard can be calculated using the formula for the area of a circle: A = πr^2, where r is the radius.
For the main board:
Radius = 8 inches
Area = π * (8)^2 = 64π square inches
For the circle with radius 6 inches (outside of the bullseye):
Radius = 6 inches
Area = π * (6)^2 = 36π square inches
For the bullseye:
Radius = 4 inches
Area = π * (4)^2 = 16π square inches
The probability of a random dart landing inside the bullseye is given by the ratio of the area of the bullseye to the total area of the dartboard:
Probability = Area of bullseye / Total area of dartboard
= 16π / (64π + 36π)
= 16π / 100π
= 16 / 100
= 0.16
Therefore, the probability of a random dart landing inside the bullseye is 0.16 or 16%.
To find the probability of a random dart landing inside the bullseye, we need to determine the ratio of the area of the bullseye to the total area of the dartboard.
1. Calculate the area of the bullseye:
The area of a circle is given by the formula A = π * r^2, where A is the area and r is the radius.
In this case, the radius of the bullseye is 4 inches, so the area can be calculated as: A_bullseye = π * (4 inches)^2.
2. Calculate the total area of the dartboard:
The total area of the dartboard includes the area of the main board and the area between the main board and the circle with a radius of 6 inches.
The area of the main board is given by: A_main_board = π * (8 inches)^2.
The area between the main board and the circle with a radius of 6 inches can be calculated as: A_ring = π * [(6 inches)^2 - (8 inches)^2].
The total area of the dartboard is the sum of A_main_board and A_ring: A_dartboard = A_main_board + A_ring.
3. Calculate the probability:
Finally, the probability of a random dart landing inside the bullseye is the ratio of A_bullseye to A_dartboard:
P = A_bullseye / A_dartboard.
Let's calculate the probability step by step:
- A_bullseye = π * (4 inches)^2.
- A_main_board = π * (8 inches)^2.
- A_ring = π * [(6 inches)^2 - (8 inches)^2].
- A_dartboard = A_main_board + A_ring.
- P = A_bullseye / A_dartboard.
Now we can substitute the values and calculate the probability.