Final answer:
To find the probability that a point chosen at random is in the blue region, we need to calculate the area of the blue region and divide it by the total area of the circle. The blue region consists of the shaded circle and the small unshaded square. The probability is equal to 1 minus the ratio of the area of the square to the area of the circle.
Step-by-step explanation:
To find the probability that a point chosen at random is in the blue region, we need to calculate the area of the blue region and divide it by the total area of the circle. The blue region consists of the shaded circle and the small unshaded square. The area of the blue region can be found by subtracting the area of the small square from the area of the circle.
First, let's calculate the area of the circle:
Area of circle: π * radius^2 = π * (diameter/2)^2 = π * (2/2)^2 = π * 1^2 = π square units
Next, let's calculate the area of the small square:
Area of square: side^2 = 2^2 = 4 square units
Now, let's calculate the area of the blue region:
Area of blue region: Area of circle - Area of square = π - 4 square units
Finally, let's calculate the probability:
Probability = Area of blue region / Area of circle = (π - 4) / π = (π/π) - (4/π) = 1 - (4/π)