Question

2. A dart is aimed at a circular board. A rectangle is drawn on the board. What is the probability that the dart hits inside the rectangle, if it is known that the dart hits the circular board.

Answer 1

Let us assume that the circular board is of radius R.
Area of the circular board = π R²
It is known that the dart will hit the circular board. Hence, the probability of hitting an area of A inside the board is = A /πR²

The size of the rectangle is not given.  So we will take the maximum rectangle that can be drawn inside the circle.  The rectangle is obviously symmetrical wrt center of circle. Let the length and width be a and b.

The diagonal of the rectangle = 2 R = √(a² + b²)
Area of rectangle A = a b = a √(4R² - a²)

differentiating and equating dA/da = 0
     =>  √(4R² - a²) -  a²/√(4R²- a²)  = 0
     =>  4 R² = 2 a²      =>  a = √2  R    and  b = √2  R

Area max = 2 R²

Probability of hitting inside the rectangle
  <= 2 R² /(π R²)
  <= 2/π

Minimum probability can be 0, if the rectangle size is very small.