We all are familiar with the Dice. Many games are available which use them. Ancient India game of Chausar, which is in the very heart of the epic Mahabharata, is an example of such a game.
Many types of dices are available. The most famous is the cubical Six Faced Dice ( here after referred as SFD). Each face is marked with a single number from 1 to 6. All faces of SFD are exactly equal to each others in dimensions.
Other types of polyhedral dices, having number of faces different from 6 can also be made, say like a polyhedral 10 faced dice.
Today I will look at the dice from a mathematical point of view. Lets explore and quantify the mathematical reasons for the results we get on rolling the dice. In the end I will conclude by touching the philosophical aspects of the ‘Dice of Life’
Now lets first consider the following situations.
1. If we throw only one SFD then the probability of getting a particular face ( marked by a particular number from 1 to 6) is exactly equal for all the six faces. And mathematically we can calculate this probability to be, 16.7 %.
2. Now lets calculate the probabilities in a slightly different situation. Here we will throw two SFDs simultaneously and add their individual face values. In this situation also the probabilities of each of the two dices, rolling to any of the six possible face values is exactly 16.7 % for each face value of each dice.
It is required in some games that we have to sum up the face values of each of the two SFD and use that sum in the play. The number of different summations values, while using two SFDs are from 2 to 12. So there are a total of 11 different summations values possible.
Suppose we make a single symmetric Eleven Faced Dice (here after referred as EFD) and throw that dice in the play, then the probabilities of getting each of the face values ( from 2 to 12) of EFD will be exactly 9 % for each face value. In other words while using a EFD the possibility of getting face value 2 will be equal to the possibility of getting face values 7 or 10 or 12, and that is 9 % for each of them.
BUT if we add the face values of two SFD, then, it is an observation that probability of getting 2 or 12 as the summation value is extremely low. The reason being that there is only 1 possible way out of 36 different possible ways in which the dices can roll, of getting the sum as either 2 or 12. And that way is, each dice getting face value as either 1 or 6 respectively.
On the contrary, the number of possible different ways in which we can get the other summation values of 3, 4, 5, 6, 7, 8, 9, 10, 11 are 2, 3, 4, 5, 6, 5, 4, 3, 2 respectively.
So, on calculations we get that, when ever we throw two SFDs, then 17 % of the time we will get 7 as the summation value, 14 % of the times we will get 6 or 8 as the summation value, 11% of the times we will get 5 or 9 as the summation value, 8 % of the times we will get 4 or 10 as the summation value, 5.5 % of the times we will get 3 or 11 as the summation value and the least i.e., 3 % of the times we will get 2 or 12 as the summation value.
Therefore while throwing two SFDs, there is highest possibility of getting a score of 7, followed by getting either 6 or 8, then still lesser 5 or 9, then 4 or 10, followed by 3 or 11 and the least possibility is of getting a score of 2 or 12.
Statistically this data will give a inverted ‘V’ shaped curve.
The beauty of the game of chance is that, when we need 7, to be in the game, then, though we have the highest, i.e., 17 % probability of getting 7 as the summation value, as compared to other values, BUT this also implies that we have 83 % probability of not getting 7 !
A deduction from the above facts is that, there are about 45 % chances ( almost 50 % ) of getting one of the following three face values i.e.,6, 7 or 8. The problem is that, at the same time, we also have 55 % chance of not getting either of 6, 7 or 8 !
3. A third situation is in which we throw three SFD and sum their face values.
Then the summation values range from 3 to 18. They are 16 in number.
Mathematically we can deduce that there are 216 different possibles permutations and combinations in which the three SFDs will roll.
By applying the same mathematics we can get that there is the highest i.e., 12.5 % probability of getting either 10 or 11 as the summation value, 11.5 % probability of getting either 9 or 12 as the summation value, 9.7 % probability of getting either 8 or 13 as the summation value, 6.9 % probability of getting 7 or 14 as the summation value, 4.6 % probability of getting either 6 or 15 as the summation value, 2.7 % probability of getting 5 or 16 as the summation value, 1.4 % probability of getting either 4 or 17 as the summation value and the least i.e., 0.5 % probability of getting either 3 or 18 as the summation value.
If we draw a graph out of this data, we will get a inverted ‘U’ shaped normal curve.
If we use a single, symmetric 16 faced dice then the probability of getting each of the face value from 3 to 18 will be 6.25 % each.
No one can be 100 % sure how the dices will roll and what face values we will get. We can only predict the probabilities of getting a particular face value, or saying the other way, the probability of not getting that particular face value.
General conclusions about results in rolling multiple dices are–
- The probability of getting same face value in all of the multiple dices is less common.
- There is a more probability of getting a middle summation value, then the extreme value.
- We can check the structural symmetry of a dice by studying the probabilities of each face value. If the probabilities are more or less equal then the dice is ‘perfectly’ made.
Philosophical aspects of the Dice of Life—
If we think philosophically, the game of dice can teach us about the truths of some aspects of life(Beware not all).
We live in our society, where rules to which we all have ‘agreed to’, decide our behaviors and the outcomes of events. The outcome’s probability can be predicted based on the basis of these rules. The various possible outcomes in a particular case can be taken as the face values of the Dice of Life.
Life doesn’t play with a ‘single’ dice, if this would be the case, then everything would become predictable and there will be no uncertainties of life, NO THRILL, NO SUSPENSE. The game of life is played with multiple dices. And the number of dices with which the life plays its game is not two, three four or five, it can be much more than that. This complicates the matter and thats what life is all about, The Complexity !
- In life, the probabilities of occurrence of extreme fortunate/unfortunate events is low. The religious people calls the occurrence of such a rare event as a ‘miracle’ and the scientists call the occurrence of such a rare event a ‘statistically rare event’.
- Most of the time mixed blend of fortunate and unfortunate events keep occurring in our life, the ‘middle summation values’.
- Every event which happens can have multiple possible outcomes, each outcome having its unique magnitude of probability of occurrence, which we can judge based upon our understanding of societal rules.The predictability of these outcomes by us depends upon our depth of understanding of these rules. We make our life decisions based on our assessment of such probabilities.
- If a person habitually overestimates the probability of occurrence of unfortunate events he is bound to be depressed. The word habitual is important here.
- If a person habitually overestimates the occurrence of fortunate events he is bound to make foolish decisions and suffer losses.
- A person with balanced and healthy mind correctly estimates the magnitude of occurrence of various probabilities. He makes life judgement based on such assessments. He doesn’t overestimate or underestimate the probabilities of occurrence of fortunate/unfortunate events. Because of thorough knowledge of the rules of the game,the error of judgement doesn’t occur. Even if he is betting his money for the rare fortunate event, he ensures that he is prepared for the occurrence of a rare unfortunate event( The Back Up Plan). And such ‘risk taking’ is a cool calculated ‘conscious’ decision, not a rash arbitrary action.
It is this understanding of the game of Dice of Life which gives a person the necessary courage to take risks, the mental strength to suffer set backs and above all, the modesty to be composed in success.!