Max has a light bulb and a defective switch. The light bulb is initially off, and on the n^{th} time the switch is flipped, the light bulb has a \frac{1}{2(n+1)^{2}} chance of changing its state (i.e. on \rightarrow off or off \rightarrow on). If Max flips the switch 100 times, find the probability the light is on at the end.