Question 160
New this year at HMNT: the exciting game of $R N G$ baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number $n$ uniformly at random from $\{0,1,2,3,4\}$. Then, the following occurs:
- If $n\gt 0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by $n$ bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point.
- If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points.
What is the expected number of points that a given team will score in this game?
- If $n\gt 0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by $n$ bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point.
- If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points.
What is the expected number of points that a given team will score in this game?