Question: Suppose that each time I throw a dart, it is uniformly distributed on a disk of radius 1 centered at the origin, and suppose the

Suppose that each time I throw a dart, it is uniformly distributed on a disk of radius 1 centered at the origin, and suppose the different throws are independent of each other.

Suppose that I keep throwing until for the first time the dart is within a distance of 1/2 from the origin.

Let S be the total sum of the distances (from the origin) for all my throws.

Compute the expectation and the variance of S.

(Please express your answers as single simplified fractions.)

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