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## Question

I am playing a dice game in which I roll a dice and add the point to one I rolled previously.

My goal is to get maximum sum but if I roll 1 I am going to loose all my points and get zero and the game is finished.

the question is following;

what should be the best strategy to get maximum sum?

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## 14 answers to this question

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roll all sixes :>)

But how is another story

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I don't think there's any strategy if I understood it right. You keep rolling the dice and adding the numbers that come up until you decide to stop, but if the number 1 comes up before you stop, you lose all your points and the game ends. Is that right? Then the probability of getting 1 is always the same: 1/6, so the probability of not getting one in n rolls is (5/6)n, but you can never know if next roll is gonna be 1, so you can't make a strategy.

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roll the dice and if a one doesnt occur after 5 rolls stop because after that probability is against you

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As you said the goal is to get the maximum score, so there are definitely some odds and strategy that come into play here. Obviously, the lower the your score is the less you have to lose so, even though you are no more likely to have a favorable result on the role, if you have a lower score when you do role a 1 you won't lose as much so the odds of the making points on the role favor you.

An Example:

That was confusing, so for example, imagine you haven't rolled yet and have 0 points. Obviously, if you roll a one you didn't lose anything so you might as well roll the dice and possibly have a chance to win some points.

Now, imagine you roll a 2 first. So your total score is 2. Rolling now there is only a 1/6 chance to roll a one. So your expected loss is only 2(your score) * 1/6 or 1/3 of a point. You have a much greater chance of rolling other numbers which are also worth more so your expected winnings is much more than 1/3, thus it is still pretty clear that you should role again.

Now some math:

In fact in every role you take your expected winning will be:

2 * 1/6 +

3 * 1/6 +

4 * 1/6 +

5 * 1/6 +

6 * 1/6 = 20/6 (or 3 and 1/3).

and your expected loss is your total score so far * 1/6 (the likelihood of rolling a one)

Thus as long as you have less than 20 points the odds are in your favor to roll again. And if you have more than 20 points the odds are against you. If you have exactly 20 points, it is 50/50 and it depends on if you like to gamble.

However, these numbers assume a pay out directly proportional to how many points you score. If you are actually playing somebody (or other factors enter the game) they might change depending on what they do.

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Nah, the best strategy in any dice game is

flick your wrist. C'mon. Dice are chance.. no strategy needed.

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I'm not clear on the rules of this "game." Is only one die involved, or more than one? Are you adding the rolls together until you reach a certain specific number? What is the goal of the game? In rolling a die, I can't see how there's any strategy -- which is why there's a saying, "luck of the roll."

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i have to agree with Ian.

payout versus risk. 20 points is the goal.

more than that and the risk is against you.

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(5/6)n = 1/2

So, up to the 3rd roll, you have more than 50% chance of not getting a 1. Rolling 4, you have a little less than 50%. So maybe rolling 4 times is a good strategy. Or combined with iangardner777 solution, 4 rolls or 20 points, whichever comes first. How's that?

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using computer analysis, i got the following.

after 100,000 trials:

stopping at exactly five rolls gives 806,790 points, more reduces, less reduces.

stopping at 20 points or more gives 810,270 points, more reduces, less reduces.

stopping at 5 rolls or 20 points gives 808,410 points. stopping when both are increased or both are decreased reduces.

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For each event, the probability of rolling a 1 on one die is 1/6.

But the question, in part, is being asked is what is the calculated probability of rolling at least one 1 in N rolls. The answer to that part of the question is 1 - (5/6)N.

The main part of the question is how many rolls can be taken before the probability of at least one 1 in N rolls exceeds 50%. This occurs after the third roll, therefore the best strategy is to stop after 3 rolls.

Roll v. Probability

[ 1 ] 1/6 ~= 0.166666667

[ 2 ] 11/36 ~= 0.305555556

[ 3 ] 91/216 ~= 0.421296296

>>>>> 1/2 = 0.5000000000 <<<<<< 50% barrier

[ 4 ] 671/1296 ~= 0.517746914

[ 5 ] 4651/7776 ~= 0.598122428

...

Edited by Dej Mar
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Haha, so in practice the different strategies work out pretty close. You would just get one extra dice roll before the house took all your money!

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As you said the goal is to get the maximum score, so there are definitely some odds and strategy that come into play here. Obviously, the lower the your score is the less you have to lose so, even though you are no more likely to have a favorable result on the role, if you have a lower score when you do role a 1 you won't lose as much so the odds of the making points on the role favor you.

An Example:

That was confusing, so for example, imagine you haven't rolled yet and have 0 points. Obviously, if you roll a one you didn't lose anything so you might as well roll the dice and possibly have a chance to win some points.

Now, imagine you roll a 2 first. So your total score is 2. Rolling now there is only a 1/6 chance to roll a one. So your expected loss is only 2(your score) * 1/6 or 1/3 of a point. You have a much greater chance of rolling other numbers which are also worth more so your expected winnings is much more than 1/3, thus it is still pretty clear that you should role again.

Now some math:

In fact in every role you take your expected winning will be:

2 * 1/6 +

3 * 1/6 +

4 * 1/6 +

5 * 1/6 +

6 * 1/6 = 20/6 (or 3 and 1/3).

and your expected loss is your total score so far * 1/6 (the likelihood of rolling a one)

Thus as long as you have less than 20 points the odds are in your favor to roll again. And if you have more than 20 points the odds are against you. If you have exactly 20 points, it is 50/50 and it depends on if you like to gamble.

However, these numbers assume a pay out directly proportional to how many points you score. If you are actually playing somebody (or other factors enter the game) they might change depending on what they do.

I like this and this was my initial reaction too.

But rolling a 1 does more damage than just losing all of your current points... it ends the game.

I believe your analysis would be absolutely correct if the game continued after you rolled a 1 and lost all of your points.

That is how you treat it in your analysis.

Here are some observations,

The probability of losing does not vary with rolls: at turn 100, you have the same probability of losing that you did at turn 1. The outcome of losing does not vary with rolls: if you fail on turn 100, you are no better off than if you failed on turn 1. From this reasoning, I believe that any long term strategy would be the same as any short term strategy, and that this strategy should not change as more rolls to the dice are changed i.e. the strategy is fixed for every turn.

To ever maximize the number of points made, the player must choose to stop somewhere. If they choose to never stop, eventually they will roll a 1 and end up with a score of 0.

At each turn here are the choices:

Stop: Keep current score. Lose potential to make more points and more decisions in the future.

Go: Good outcome, increase points, can still make more rolls (5/6) and Bad outcome, lose all points, can no longer make any rolls (1/6)

To keep any good outcome, the player will eventually need to choose stop. Any "goodness" tied to the good outcome is essentially tied to chance beforehand, and choosing stop.

Ultimately, the game will end with either a player choosing to stop or rolling a 1.

Stop is clearly the better outcome whenever you have more than 0 points.

The only time choosing to roll is truly justified is the first turn. After the first turn, I would choose to stop.

Here is a more clear statement of the reason.

There is no strategic difference in choosing one action over another for any turn after turn 1.

The only way to win the game with non-zero points is to eventually choose to stop. The winning strategy is therefore to choose the right time to stop. Since the strategy for every turn after turn 1 is the same, and must include stopping, the entire strategy must be to stop after turn 1. Turn 1 is a special case, because choosing to roll leads the player to be better off or just the same as if he chose not to roll (assuming no fee for playing this game).

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It depends on how long you want to play. If the strategy is to end the game then you want to roll a '1'. In this game to end it you have to roll a '1', and if you roll a '1' you lose all your points so there is no point in keeping score because eventually "you will roll a '1'".

If the strategy is to see how long you will last (time) then the only points you are accumulating is the number of times you roll the die or how long you last in the game because all die totals will be removed in the end.

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It depends on how long you want to play. If the strategy is to end the game then you want to roll a '1'. In this game to end it you have to roll a '1', and if you roll a '1' you lose all your points so there is no point in keeping score because eventually "you will roll a '1'".

If the strategy is to see how long you will last (time) then the only points you are accumulating is the number of times you roll the die or how long you last in the game because all die totals will be removed in the end.

I thought the strategy was to maximize your score. The choices I thought were to roll or quit the game with the current score.

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