2. Estimating π. Suppose we had forgotten the formula for computing the area of a circle. All is not lost. We can easily use the computer to approximate the circle’s area by simulation. For convenience, we want to determine the area of the unit circle with radius 1. We name this unknown constant Pi (or π). We know that the area covered by a square with the corner points (−1, 1), (1, 1), (1, −1) and (−1, −1) equals 4, as the length of each side is 2. This square completely contains the unit circle. If we had a random point somewhere in the square, we could calculate the probability that the point is also inside the unit circle. A random point Z is determined by two coordinates, say X and Y , which have independent uniform distributions on the interval [−1, 1]. The probability of Z being inside the circle is given by the ratio of the circle’s area and the square’s area or, more mathematically, letting P = probability: P(Z is inside the circle) = (area of the circle)/(area of the rectangle) = Pi/4. Knowing this, we can estimate (the unknown) Pi