What is the probability that the sum of 10 fair dice is divisible by 6?
Suppose we roll 10 fair standard dice. What is the probability that their sum is divisible by 6, assuming that the rolls are independent.
Calculating the probability that the sum of 10 fair dice is divisible by 6 involves a fair bit of combinatorics. Since each die has 6 sides, there are 6^10 possible outcomes when rolling 10 dice. We will use generating functions to find the number of outcomes with sums divisible by 6.
A generating function is a polynomial that represents the number of ways to get each outcome in a problem. For a single die, the generating function is:
x^1 + x^2 + x^3 + x^4 + x^5 + x^6
To find the generating function for the sum of 10 dice, we raise the single die generating function to the 10th power:
(x^1 + x^2 + x^3 + x^4 + x^5 + x^6)^10
Now, we need to find the coefficients of the terms in the expanded polynomial that are divisible by 6. These coefficients represent the number of outcomes with sums divisible by 6.
The minimum sum we can obtain from 10 dice is 10 (all ones), and the maximum sum is 60 (all sixes). We are interested in the coefficients of x^k in the expanded polynomial, where k is a multiple of 6 in the range of 10 to 60 (12, 18, 24, 30, 36, 42, 48, 54, and 60).
Expanding and finding the coefficients of the terms divisible by 6 by hand is impractical. However, you can use a computer algebra system (CAS) or programming language to do this task efficiently.
Once you have the coefficients for the terms divisible by 6, sum them up to get the total number of outcomes with sums divisible by 6. Finally, divide this sum by the total number of possible outcomes (6^10) to get the probability:
P(sum divisible by 6) = (number of outcomes with sums divisible by 6) / (6^10)
Keep in mind that this approach requires a CAS or programming language to perform the calculations.