Simulation of a Pair of Dice

A dice has numbers ranging from between 1 and 6. Two dice when cast together yield nearly 36 possibilities. Simulation is used to study random phenomenon by replicating the real world with sufficient certainty. For this let us begin with enumerating the possibilities using a pair of dice. There are 36 possible combinations which yield a sum between 2 and 12 on both faces. For example the probability of getting a 2 is very low which is only 1/36. Similarly the probability of getting 3 is (1,2) (2,1) which is 2/36 or 1/18. The chance that we might get a sum of 4 is (1,3) (2,2) (3,1) or the probability of getting a sum of 4 with two dice cast is 4/36 or 1/9.

(2,1,0.027777778,0.027777778), (3,2,0.055555556,0.083333333 ), (4, 3, 0.083333333, 0.166666667), (5,4 0.111111111, 0.277777778), (6, 5 0.138888889, 0.416666667), (7, 6 0.166666667,0.583333333), (8,5 0.138888889, 0.722222222), (9,4 0.111111111, 0.833333333), (10, 3,0.083333333, 0.916666667), (11, 2, 0.055555556,0.972222222), (12, 1, 0.027777778, 1)

Shown above is the number of occurrences of a combination on two dice. The sum of the two numbers range between 2 and 12, along with it the number of possible combinations of sums that can occur, the next shown is the probability and the cumulative probability.

Now one can go ahead and generate a large number of random numbers between 0 and 1. These would also be uniform random variates. Based on which random number occurs the corresponding range is determined and the sum assigned to that range is determined. For example if one generates a random number 0.67 then it occurs between the cumulative probability ranges 0.5833 and 0.722. The corresponding sum assigned to this cum probability range is 7. Likewise thousands of random numbers have to be generated and iterations done many times in order to obtain a set of perfectly normally generated random numbers. This is also ascertained by the central limit theorm.

When this whole model of two dice being cast was simulated over 1000 random trials and 1 iteration, s the result did not converge to the expected probability of occurrence. The simulated result shows that Three has occurred only once, Seven has occurred five times, Five has occurred 4 times, four has occurred 3 times which is different from the number of occurrences and expected probability of occurrence. This is due to the fact that one may have used a flawed RndNum generator and also due to the fact that the dice pair system has not been simulated for a sufficient number of trials and replications. As one thousand random numbers cannot be pasted in this tutorial, this has been skipped. Doing so will assist in the RndNumbers converging to a normal distribution and will also aid in computing the occurrences of the totals in the simulated system.