Let there be a tiger, William, at the origin. William leaps 1 unit in a random direction, then leaps 2 units in a random direction, and so forth until he leaps 15 units in a random direction to celebrate PUMaC’s $15$th year.
There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly \frac{1}{2} after the 15 leaps. The area of that circle can be written as A \pi. What is A?