Maths problem: fatigue vs critical hits

[lch]

cleveland said:
Here's roughly what I did:
1) Constructed a vector A(i) of the explicit probabilities for each single drn outcome i. This is pretty straightforward:
A(1)=A(2)=...=A(5)=1/6
A(6)=A(7)=...=A(10)=(1/6)^2
Etc.

I think this is already where I fail, because the rest is pretty much the same. I was jumping back and forth between modulo 5 and modulo 6 just because the whole thing is offset by 1 and the gaps didn't work out. Let's correct this now...

Let X-1 = 5a + b, 0 <= b < 5. (notice the -1!!) Then the probability that a DRN gives exactly X is p(DRN = X) = (1/6)^(a+1).
The probability that it is X or greater is:


Now, we have two ways of measuring all the (x,y) in IN x IN which sum is equal to or greater than X: Either we take all the (x,y) with x+y < X and subtract the probability of them showing up simultaneously from 1, thus inverting the set, like you did if I understood right, or measure the following (with a loose mathematical notation): Code:

---

  (1, >= X-1)

  (2, >= X-2)

  (3, >= X-3)

   ... etc.

  (X-1, >= 1)

  (>= X, *)

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which should be equal to {(x,y) | x+y >= X}. I implemented both versions in Python again: Code:

---

#!/usr/bin/env python



# p(DRN) >= x

def p_1(x):

	if x < 1:

		return 1

	b = (x-1) % 5

	a = (x-1) / 5

	return (6-b)*pow((float(1)/6), a+1)



# p(DRN) == x

def p_2(x):

	if x < 1:

		return 0

	b = (x-1) % 5

	a = (x-1) / 5

	return pow((float(1)/6), a+1)



# p(2d6oe) >= x

def doubledrn(x):

	M = [(a,b) for a in range(1,x-1) for b in range(1,x-a)]

	s = 0.0

	for (x_1, x_2) in M:

		s += p_2(x_1)*p_2(x_2)

	return 1-s



# p(2d6oe) >= x (alternate version)

def doubledrn_2(x):

	s = p_1(x)

	for y in range(1,x):

		s += p_2(y)*p_1(x-y)

	return s



for x in range(2,20):

	print ">= %2d : %f" % (x, doubledrn(x))

	print ">= %2d : %f" % (x, doubledrn_2(x))

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and huzzah, they both give the same result, and it is: Code:

---

>=  2 : 1.000000

>=  3 : 0.972222

>=  4 : 0.916667

>=  5 : 0.833333

>=  6 : 0.722222

>=  7 : 0.583333

>=  8 : 0.462963

>=  9 : 0.361111

>= 10 : 0.277778

>= 11 : 0.212963

>= 12 : 0.166667

>= 13 : 0.127315

>= 14 : 0.094907

>= 15 : 0.069444

>= 16 : 0.050926

>= 17 : 0.039352

>= 18 : 0.029578

>= 19 : 0.021605

---