euler205.py

import itertools

def map_sums_to_throws(dice,sides):
	x = list(range(1,sides+1))
	d = {}
	l = [p for p in itertools.product(x, repeat=dice)]
	for i in l:
		s = sum(i)
		if s not in d:
			d[s]=1
		else:
			d[s]+=1
	return d


sums9d4 = map_sums_to_throws(9,4)
sums6d6 = map_sums_to_throws(6,6)


## accum throws that sum up to i
accum6d6 = {6:1}
for i in range(6+1,(6*6)+1):
	accum6d6[i]=sums6d6[i]+accum6d6[i-1]

## prob sum 9d4 equals i times sum 6d6 less than i 
suma = 0
for i in range(9,9*4+1):
	p9d4is_i,p6d6less_i = (sums9d4[i]/4**9),(accum6d6[i-1]/6**6)
	suma+= p9d4is_i*p6d6less_i

print(round(suma,7))