Dice game statistics

Factual Questions
1

I apologize for the wordiness of the question in advance but it requires some explanation. I have a question related to a dice game called 1000 or Ten Thousand (depending to whom you speak).

I’m sure the rules are a little different depending on where you learned it also, but this is actually a statistical question so it shouldn’t make a whole lot of difference but here are the basics:

When the first player reaches the required amount of points to win, there is one more opportunity for each of the remaining players to see if they can beat that score.

6 dice are rolled. The scoring dice are kept and the others re-rolled. If, in the end, all of the dice are scoring then you can roll all 6 dice again. This can continue indefinitely so that, technically, there is a limitless amount of points that could be scored on an individual’s turn.

So here is the scenario:

Player A has scored the required amount of points.
Player B is taking his or her last turn and has rolled 4 scoring dice.
Player B does not have enough points to possibly beat Player A without getting 6 scoring dice so that he or she can re-roll all 6 dice and continue.
The only amounts on the dice that can score are 5 and 1.
Player B rolls a 3 and a 5.

The argument was about the next roll. One side states that since Player B must roll 6 scoring dice to have any chance at getting a higher score than Player A then they would be best served by re-rolling just the 3 again since it would then be a 2 in 6 chance of being able to continue.

The other side of the argument states that it is best to re-roll both the 3 and the 5 again since you then have an additional opportunity of rolling a combination of a 1 and 5, and if that is not achieved still have a likely chance of getting a 1 OR a 5, thereby putting you back in the same position your were in before rolling both dice.

Keep in mind that if Player B re-rolls both dice and gets neither a 5 or a 1, he or she is unable to continue.

I just don’t know enough about statistics to determine which side is correct.

Statistically, which is more likely to result in Player B’s ability to move on and roll all 6 dice again, re-rolling just the 3, or the 3 and the 5?

Thanks.

2

Have I got this right?

Player B needs to score either a 1 or a 5 on a d6 to score.
He has 4 scoring dice already, but needs two more (i.e. both the remaining ones).

Assuming that’s right then there are two possibilities:

Player B keeps the 5 and rerolls the 3. He needs either a 1 or a 5, which is 2 in 6 = 1 in 3 = 33.3%.

If he rerolls both dice, then there are 36 possibilities (1 to 6 on each dice). Of these, the only successes are:

1 1
1 5
5 1
5 5

That is 4 in 36 = 1 in 9 = 11.1%

So Player B is 3 times more likely to win if he keeps the 5.

3

I like that game (Farkle?). If you roll a 5 and 3 though, don’t you have to keep the five or lose?

Assuming you reworded it so it would work, you are better off keeping the 5 and trying with one die.

Odds of getting a 1 or a 5 on 1 roll are 1/3 or 33.3%

If you roll two dice you have a 1/9 chance of getting two 1s or 5s, a 4/9 chance of getting one 1 or 5 (which turns into a 4/27 chance of also getting a 1 or 5 on the next role), and a 4/9 chance of getting neither. So the odds of being successful this way are 1/9 + 4/27 or just under 26%.

4

If you get a 1 or a 5, you get to reroll the other die. so 5, 3 followed by a 1 would also be a successful roll.

5

that is correct.

6

Thanks hawkeyeop. I didn’t really even know how to begin going about calculating the odds for that. I was actually wondering about the ability of being able to re-roll that die, but it was my first time playing and that’s how the rules were explained (only one out of the 6 of us had played before).

7

You are welcome. If you change your scenario to rolling 3 dice instead of two and getting a 5,5,3, your question would work. You would have to keep at least one five, but you could use this to figure out if you should keep both.

8

I couldn’t come up with the odds, but my basic feeling was that if the extra roll affords no more than a 50/50 opportunity to proceed (getting a 1 or a 5) and less than a 1 in 3 chance of re-rolling all 6 dice (getting a combination of 1’s and/or 5’s), it would be to your detriment to re-roll both dice since you currently have better odds than that rolling the one die.

I couldn’t explain away the possible cumulative effect of having an extra roll on the odds though.

9

OK, looking at it again, I think I’m missing something. When you wrote “which turns into a 4/27 chance of also getting a 1 or 5 on the next role”, wouldn’t it still be a 33% chance of getting a 1/3 on the next roll?

10

1/9: success on one roll

(4/9 * 1/3) = 4/27: success on two rolls

4/9: failure on one roll

(4/9 * 2/3) = 8/27: failure on two rolls

Make sense?