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Roll of the die (Posted on 2005-04-21) Difficulty: 4 of 5
The object of the dice game is to be the first player to reach a score of at least 100 points.
Each player’s turn consists of repeatedly rolling a die.
After each roll, the player has two choices: roll again, or stop.
- If the player rolls 1, nothing is scored in that turn and it becomes the opponent’s turn.
- If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues.
- If the player stops, the turn total (the sum of the rolls during the turn), is added to the player’s score, and it becomes the opponent’s turn.

What's your strategy?

See The Solution Submitted by Hugo    
Rating: 4.1111 (9 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: a wiser approach | Comment 10 of 23 |
(In reply to a wiser approach by armando)

You are correct, Armando, that in the absence of competition, and when you are not close to 100, you want to maximize the score on a roll. 

However, you are not required to decide in advance how many times you will roll, and it would be bad strategy to do so.  Consider a player who rolls a total of 12 (6 twos) on his first 6 rolls.  He should make the exact same decision on whether to roll again as should the player who only took 2 rolls to get to 12.  It does not make sense in the first case to quit when you have 12 points, and in the second case to roll 4 more times. 

In the absence of competition, and when you are not close to 100, I assert that the decision on whether to "hit" or "stand" should be based on how many points you have for the turn (since you risk them all with your next roll), as compared to how many you expect to have for the turn if you roll again.

If you have already rolled 21 points, then after rolling again you will have (with equal probability), either 23, 24, 25, 26, 27 or 0 for the turn.  Expected value = 20 + 5/6, which is less than 21, so you should take your 21 points and pass the die.

If you have already rolled 19 points, then after rolling again you will have (with equal probability), either 21, 22, 23, 24, 25 or 0 for the turn.  Expected value = 19 + 1/6, which is more than 19, so you should take another roll.

Note that the number of rolls it took you to get to 19 or 21 does not factor in at all.
 

Edited on April 23, 2005, 4:31 pm

Edited on April 23, 2005, 4:33 pm
  Posted by Steve Herman on 2005-04-23 16:30:05

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