Mahabharata: How fair were Shakuni’s dice — A statistical approach
Mahabharata is an ancient Indian epic that narrates a story of an establishment known as Hastinapur. The focus of the story is the two groups of cousins (Kauravas and Pandavas)who ultimately went for war against each other. The story beautifully describes the development of each character and the dilemmas of right and wrong around them. The story has been given the shape of a Television series by a very famous Indian producer and director called B. R. Chopra, the episodes of which are still available on YouTube.
This article revolves around a very popular episode of the same series where the cousins play the game of Chausar which later on become the reason for the great war of Kurukshetra. This article is based on a TV episode and may not be compared with the literature in books. The part of the episode where the game is played is pasted below:
Chausar is a game much similar to Ludo and typically played by two or three stick dice as shown below:
Some dies have the numbers of 1, 3, 4, 6 on their wider faces while the others can have 1, 2, 5, 6 on them. Which type of stick dies were exactly used in the game, I don’t know but let’s assume they used one from each type.
In this game of Mahabharata, they used to throw a pair of dice and the numbers that show up will decide the steps the pawn can move (as I said much similar to the game of Ludo). As an example, if the pair is thrown and one die shows 4 and another shows 2, the pawn will move 6 steps (sum of the two numbers). Also, before throwing the dice set, the person predicts a number and if that number comes out to be true he gets another chance.
This second rule of prediction and getting another chance has an important role in the game played in Mahabharta. On one side we had Kauravas represented by a Kaurava brother called Duryodhana. Duryodhana is assisted by his maternal uncle called Shakuni. It is said that Shakuni had a special pair of dice with him that obeyed his command.
On the other side were seated Pandava brothers who were represented by their elder brother called Yudhishthira. The game was started by Kaurava brothers and as I said they had Shakuni on their side and because he had a special dice set with him Pandavas never got a chance to play. This ultimately lead Kauravas and Shakuni to win everything Pandavas put on Stake including their wife, Draupadi. The embarrassing defeat and the subsequent events lead to the war of Kurukshetra.
In this article, we will put a statistical check on Shakuni’s magical dice, whether they actually obeyed his command or was it just a chance that lead Kauravas to win the game. We will see the probability associated with the game outcomes and how possible it was for Kauravas to win.
Before jumping for calculations, let us assume the pair of dice that were used in the game had one die of [1, 3, 4, 6] and another [1, 2, 5, 6]. So if Shakuni threw the dice and predicted the number 8, it will be possible if he gets 3 on Die 1 and 5 on Die 2. Another possibility is 6 on Die 1 and 2 on Die 2. We can calculate the probability of this event keeping these numbers in mind as shown below:
The probability of getting 3 on Die 1 and 5 on Die 2 is (1/4) * (1/4) = 0.0625
Also, the probability of getting 6 on Die 1 and 2 on Die 2 is (1/4) * (1/4) = 0.0625
Hence the probabilty of getting number 8 will be 0.0625 + 0.0625 = 0.125 or 12.5%
Suppose after number 8 comes true, Shakuni bets on the number 12. The probability of this event seen as separate from the first one would be 0.0625 (calculated in a similar way as that of number 8). But since these were a series of independent events (statistically speaking) the probability of getting 8 in the first throw and 12 in the 2nd would be the multiplication of two probabilities i.e. 0.125 * 0.0625 = 0.00783
The Game that demanded war
Now coming to the game where Pandavas and Kauravas were playing against each other. The Shakuni on the commands of Duryodhana started the game and played it in such a way that his every prediction turned out to be true. The things that Yudhishthira put at stake and ultimately lost were:
Necklace, 118000 Gold Coins+ treasure, Yudhishthra’s Chariot, etc.
We will skip the part involving necklace, not because the necklace seems cheap when compared to other things put on the bet but it involved a multiple number of dice throws and can involve a separate calculation, and honestly, it steals the excitement of results.
After winning the Necklace, the betting process and the game goes like as shown in the below table:
The numbers listed under the column of Dice count were predicted by Shakuni and the associated probabilities of those numbers are shown in the Probability column. The cumulative probability means the probability of getting this number and all proceeding numbers e.g. the number 6 in the third row has the cumulative probability of 0.048%, which means getting 12 in the first throw, 8 in second, and 6 in third has the probability of 0.048%.
All the numbers listed in the Dice count column were predicted by Shakuni and the prediction turned out to be true every time.
If we analyze the above probability values, it becomes clear how the probability of events drastically dropped after the third throw. Let us do it in a more statistical manner. Let’s define our null and alternate hypothesis:
Null Hypothesis: The dice set was fair or the dice were obeying the command of Statistics
Alternate Hypothesis: The dice set was not fair or the dice were obeying the command of their lord (Shakuni)
Let’s set our level of significance to be 0.05% (very least to lower our expectations for this case)
From the above table and graph, it is clear that we should have rejected the null after the third throw, where probability was just 0.048% (less than the level of significance).
So Statistically speaking Yudhishthira must have stopped the game when he lost One Lakh maidservants including their ornaments. The event that sparked the most outrage was putting Draupadi on the bet and losing her, the probability of which is almost zero even when expanded to 10 decimal places
So the final verdict is: Yes, Shakuni’s dice listened to him
Thanks for reading the article, suggestions are welcome