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[Native English speakers point here for note]
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Quote: The die is cast
(Gaius Julius Ceasar)
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The Game of Dice
Contents
A dice is a small cube that has small marks
cast on it. Six sides of dice are labeled with
1 to 6 marks. This made dice an ideal tool for ruining people.
On the other hand it had also boosted initial interest into probability.
The set of six dice - usually called the dice game - is
experimental gem that allows endeavors in pure experimental
probability calculations.
Dice is one of the oldest games, played even by ancient Romans
as shown in the above quote by Julius Caesar.
In history lots of people lost and won lots of property by playing
this game. It was, it is and it will be popular because it is
like life - you have to roll again and again dreaming about
a better score even when you know that there could be a zero
score after the next throw and that ultimately there will be a
zero at the end if you don't stop rolling.
Dice is risks based but contrary to the other games,
like roulette, players have a big chance to influence their
resulting score by evaluating the risk of the given position.
The analysis of the optimal strategy is amusing task for the
investigation.
I. Beginning of the game:
Two or more players play the dice. Before game players have agreed on:
- the final score (usually 10,000) reaching which means victory in the game
- the minimal turn score (usually 350) i.e. minimal score that allows to end turn
Also some additional counting rules could be agreed at the beginning of the game (see point IV).
The game starts by choosing player who will start round. The selection is usually based on the highest face on a die thrown by each player.
II. Rules for counting of the player score during the turn
Player starts round throwing full set of six dice. Player throws and search for any of following recognized countable combinations of dice:
- Faces of the dice set consist from exactly 1,2,3,4,5 and 6 points (called sequence - of course valid only for throw with full set of six dice) - add 2000.
- Three dice with face equal one - add 1000.
- Three dice with the same faces - add 100 times face
- Four (five,six) dice with the same face - add 200 (400,800) times of the face value or 2000 (4000,8000) for face equal to one.
- Dice with face equal to one - add 100 for each each
- Dice with face equal to five - add 50 for each
If player's throw results in any countable combination (or combinations), its (their) score(s) is (are) added to the player's turn score. Player has to give aside all dices that were used for increasing turn score. The remaining dice set is used for the next throw. If there is no remaining dice (all dice were used for increasing player turn score), the player has use again full set of six dice in the next throw.
If player's round score exceeds agreed minimal turn score limit player can stop throwing and his/her current turn score is added to his/her overall score. The next player starts his/her turn in the round.
If there is no recognized dice set, player's turn score is set to zero his/her overall score is left unchanged and dice are pass to the next player. Otherwise, as far as there is any of the scoring combination listed above, player can continue turn by throwing non-scoring dice (or using again whole dice set if all dice were used for score). It is his/her purely decision when turn is closed and current turn score is added to player's score.
III. Determining the winner:
The overall players' score are compared after each round. Winning order of the players is given by:
- Order in which players score reached (or exceeded) agreed final score limit
- In the case more players reached the final score limit in the same round, order is determined by players' score.
IV. Optional constraints for increasing risk:
There are several modifications of the above rules that generally make the dice game more risky (and its analysis nearly impossible):
- Player is not allowed to stop turn when his/her next throw would use full set of six dice.
- Minimal one round score limit could be changed according to player's overall score (i.e. increased from 350 to 500 when player's score is higher than 5000).
- The final score should be reached exactly (obviously in this case there is no minimal round's score for the last round)
- Player does not need to give aside all scoring dice, it is allowed to give away (and count) just one scoring combination and then continue throwing (with the higher number of dice).
- Player could start round by continuing with the position (and score) that was quit by previous player
There is no analytical way how to calculate average score achieved when
one throws I dice. Fortunately it is simple to evaluate all combination with just 6 dice sets.
Program (in SAS ) for doing it is stored HERE.
The resulting Table 1 is presented bellow.
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Table 1: Average scores and Probabilities for single throw |
I=Initial
number
of dice |
K= next
size of
dice set |
K=1 |
K=2 |
K=3 |
K=4 |
K=5 |
K=6 |
K=0 |
(K=1..6) |
| 1 |
P(I,K) [%] |
|
|
|
|
|
33.33% |
66.67% |
33.33% |
| |
S(I,K) |
|
|
|
|
|
75.00 |
|
75.00 |
| 2 |
P(I,K) [%] |
44.44% |
|
|
|
|
11.11% |
44.44% |
55.56% |
| |
S(I,K) |
75.00 |
|
|
|
|
150.00 |
|
90.00 |
| 3 |
P(I,K) [%] |
22.22% |
44.44% |
|
|
|
5.56% |
27.78% |
72.22% |
| |
S(I,K) |
150.00 |
75.00 |
|
|
|
362.50 |
|
120.19 |
| 4 |
P(I,K) [%] |
13.58% |
29.63% |
37.04% |
|
|
4.01% |
15.74% |
84.26% |
| |
S(I,K) |
361.36 |
150.00 |
75.00 |
|
|
553.85 |
|
170.33 |
| 5 |
P(I,K) [%] |
11.06% |
21.09% |
30.86% |
26.23% |
|
3.03% |
7.72% |
92.28% |
| |
S(I,K) |
568.60 |
360.37 |
150.00 |
75.00 |
|
689.62 |
|
244.67 |
| 6 |
P(I,K) [%] |
9.52% |
18.65% |
24.69% |
24.69% |
15.43% |
3.94% |
3.09% |
96.91% |
| |
S(I,K) |
713.51 |
584.48 |
359.38 |
150.00 |
75.00 |
1,323.53 |
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377.99 |
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Average scores are valid for one throw
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i.e. not corrected for scores from subsequent throws |
How to read it?
When one throw 5 dices there is probability 7.72% chance of bad luck
(positions without scoring dice combination). On average score will be
increased by 244.67, but there are different probabilities
of ending with 1 to 4 dices each of which has its own average score
e.g. we have 21.09% chance to continue throwing with 2 dices,
in which case the average score increase will be 360.37.
Table 1 lists average scores increase after one throw - it does not take into account additional score increments from potential subsequent scores.
Now we will analyze final score when we will start with set of I dice and we will continue throwing as far as there will be any scoring combination. (Imagine the player who never stops turn - and we are just recording player's turn score before the moment he/she roll zero).
Let:
P(I,K) is probability of that next throw with K dice, when we throw I dice (see table 1)
S(I,K) is average score from single throw of I dice resulting to next throw with K dice (see table 1)
W(I) is average score achieved when rolling I dices until zero score is reached
Following equations describe relation between P(I,K) S(I,K) and W(I):
W(1) = P(1,6).[W(6)+S(1,6)]
W(2) = P(2,6).[W(6)+S(2,6)] + P(2,1). [W(1)+S(2,1)]
W(3) = P(3,6).[W(6)+S(3,6)] + P(3,2).[W(2)+S(3,2)] + P(3,1).[W(1)+S(3,1)]
...
W(6) = P(6,6).[W(6)+S(6,6)] + P(6,5).[W(5)+S(6,5)] + P(6,4).[W(4)+S(6,4)] +
+ P(6,3).[W(3)+S(6,3)] + P(6,2).[W(2)+S(6,2)] + P(6,1).[W(1)+S(6,1)]
The SAS/IML program that solves above equations is listed HERE.
SAS/IML output explains well how the solution is constructed.
There is simple relation between average potential score W(I)
and decision when stop throwing. If we have K dice set, the current
turn score C than the next average current turn score C' is equal:
C' = [C + W(I)] .[1- P(I,0)]
Where P(I,0) is probability of zero score when throwing I dice
It makes sense to continue only if average score increase achieved
by the next throw is positive:
C' - C = [C + W(I)] . [1- P(I,0)] - C = - C . P(I,0) + W(I) .[1- P(I,0)] >0
Critical value of current turn score is equal:
Cs(I) = W(I) .[1- P(I,0)] / P(I,0)
The table 2 lists critical scores for all sizes of the initial dice set.
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Table 2: W(I) - Average scores accumulated until throw |
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with zero score and critical scores Cs(I) for stopping |
I=Initial
number
of dice |
W(I)=Average
score gain |
Cs(I)=critical
score for
stopping |
| 1 |
263.37 |
131.68 |
| 2 |
246.51 |
308.14 |
| 3 |
294.62 |
766.01 |
| 4 |
390.13 |
2,088.37 |
| 5 |
521.89 |
6,241.85 |
| 6 |
715.10 |
22,454.28 |
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Stop throwing if your current score is above Cs(I) |
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and you are deciding with I dice in hand |
The decision about the next throw with I dice is simple -
if the current turn score C is above value of Cs(I) listed in Table 2
the risk of zero score is higher that potential score gains - so it is wise to stop throwing.
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