What is the total, between 12 and 72, of rolling a dozen dice that will give the greatest probability of occurrence? Secondly, what is that probability?
Thank you very much for help.
The first part is easy enough. Will simply take the average of the 12 dice as follows:[1 + 2 + 3 + 4 + 5 + 6=21/6 =3.5 x 12 =42 - this is the most likely total for rolling 12 dice, or the expected total.]
Now, the second part, however, is much more difficult to calculate. There is a rather complicated formula that will calculate the total, and it comes from this page of "Mathword": http://mathworld.wolfram.com/Dice.html {Equations 7, 8, 9, 10}, where s = number of sides on each die = 6, n = number of dice = 12, p = total of dice rolled = 42, k=number of summations =[ (42 - 12)/6] =0 to 5.
∑(-1)^k*bin(12, k)*bin[(42 - 6k - 1), (12 -1)], k=0 to [(42 - 12)/6]
I entered the above into my computer and it summed them up pretty quickly: = 144,840,476 / 6^12 =0.0665387960.......etc., which is the probability of a total of 42.
Note: The above number of 144,840,476 can also be obtained, as the coefficient of x^42 in the expansion of this polynomial: [x + x^2 + x^3 + x^4 + x^5 + x^6]^12.
