# Throwing a Die to Score 100 Points

Throwing a Die to Score 100 Points

A score of 100 points wins a certain dice game. Each turn, a player rolls a single die as many times as they like, and gains either zero points if a 1 is ever rolled or else the total of their rolls. After a turn, the score for the turn is added to their total score. Other players are also competing to reach 100 points first.

For example, one of the players could get, in one turn, points 5, 3, 5, 2, and 6 (she decides to stop here), for a score of 21. This is added to her total score. Another player could get the points 4, 3, 5, and, unfortunately, 1, so that she must stop throwing the die and gets a score of 0 for the turn in question.

We apply the following strategy for the game. At each turn, the player tries to get a given threshold value, for example, 20. Once her score is at least this value, she stops throwing the die for that turn.

The Demonstration shows four kinds of results:

1) sample paths up to 100 points

2) histograms of the number of turns required to get 100 points

3) the average number of turns required to get 100 points for each threshold value

4) the optimal strategy that maximizes the expected score in a turn