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(Assume all throws fall inside a circle with radius r = 1)
- AingeruKatua May 20, 2015With uniform probability, all darts can fall anywhere on the area of the unit circle with the same probability. Think of the problem as a monte-carlo integration problem, where after 1000s of simulations (dart throws). We can split the area of the circle in two equal halfs using a concentric inner circle of radius e (where intuitively r > e > 0.5r). In this case, e would be the expected average payoff.
A = PI * r^2
0.5 A = PI * e^2
r = 1
e = sqrt(0.5) = 0.707107