An individual named Claudius is located at the point 0 in the accompanying diagram....

An individual named Claudius is located at the point 0 in the accompanying diagram.

Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius first moves to one of the four locations B₁, B₂, B₃, B₄. Once at one of these locations, another randomization device is used to decide whether Claudius next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin.

a. Let X = the number of moves that Claudius makes before first returning to 0. What are possible values of X? Is X discrete or continuous?

b. If moves are allowed also along the diagonal paths connecting 0 to A₁, A₂, A₃, and A₄, respectively, answer the questions in part (a).