Suppose you want to know the probability of rolling a 10. To find the probability on 3D6, divide the coefficient for the term with degree 10 by 6 to the power 3 because we rolled 3 D6s. To find the probability on 4k3, divide the coefficient for the term of degree 10 by 6 to the power 4 because we rolled 4 D6s.
To find the probability of rolling lesser than 10, add the coefficients of the terms of degree < 10 and divide. To see the probability of rolling greater than 10, do the same with terms of degree > 10.
3D6 polynomial
Code: Select all
x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 + 21x^8 + 25x^9 + 27x^10 + 27x^11 + 25x^12 + 21x^13 + 15x^14 + 10x^15 + 6x^16 + 3x^17 + x^18
ROLL | PREDICTION | RESULT | RESULT
Roll < 10 | 3/8 = 37.5% | 37.23% | 37.12%
Roll = 10 | 1/8 = 12.5% | 12.97% | 11.93%
Roll > 10 | 1/2 = 50.0% | 49.90% | 50.95%4k3 polynomial
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x^3 + 4x^4 + 10x^5 + 21x^6 + 38x^7 + 62x^8 + 91x^9 + 122x^10 + 148x^11 + 167x^12 + 172x^13 + 160x^14 + 131x^15 + 94x^16 + 54x^17 + 21x^18
ROLL | PREDICTION | RESULT | RESULT
Roll < 10 | 227/1296 = 17.52% | 17.52% | 17.29%
Roll = 10 | 61/648 = 9.41% | 9.25% | 9.52%
Roll > 10 | 947/1296 = 73.07% | 73.46% | 73.80%For each method, only one way to roll a 3 exists. For the 3d6 method, only one way exists to roll an 18; however, 21 ways exist for the 4k3 method.