Π appears in the formula for the standard normal distribution, the most important probability distribution in statistics. Why not give it a try to calculate π using statistics! In fact, you'll use a simulation technique called the Monte Carlo Method.

Recall that the area of a circle of radius r is A = πr2. Therefore the area of a circle of radius 1, aka a unit circle, is π. You'll compute an approximation to the area of this circle using the Monte Carlo Method.

a) The Monte Carlo Method uses random numbers to simulate some process. Here the process is throwing darts at a square. Assume the darts are uniformly distributed over the square. Imagine a unit circle enclosed by a square whose sides are of length 2. Set an R variable area.square to be the area of a square whose sides are of length 2.

b) The points of the square can be given x-y coordinates. Let both x and y range from -1 to +1 so that the square is centred on the origin of the coordinate system. Throw some darts at the square by generating random numeric vectors x and y, each of length N = 10,000. Set R variables x and y each to be uniformly distributed random numbers in the range -1 to +1. (hint: runif() generates random number for the uniform distribution)

c) Now count how many darts landed inside the unit circle. Recall that a point is inside the unit circle when x2 + y2 < 1. Save the result of sucessfull hits in a variable named hit. (hint: a for loop over the length of x and y is one option to reach hit)

d) The probability that a dart hits inside the circle is proportional to the ratio of the area of the circle to the area of the square. Use this fact to calculate an approximation to Π and print the result.