By comments after adding mpmath package to python one can run this program to see the result:
from mpmath import *
mp.dps=50
def th(x):
if x<0:
return 0
else:
return 1
def f(r,k):
a=(-1)**k*binomial(100,k)*binomial(r-10001*k+99,99)*th(r-10001*k+99)
a= a/binomial(r+99,r)
return a
from timeit import default_timer as clock
time0 = clock()
s=0
for r in range(493701,506200):
for k in range(0,101):
s=s+ f(r,k)
time1 = clock()
print 'The result is'
print s
print 'computed in ', round(time1-time0,3), 'sec'
"""
Result:
0.0000000013060480365934635751811931267930306865307928
"""
mpmath is available in Ubuntu repository. Also one can write a C++ program using GMP: http://rextester.com/TGS53335 (this can be run only in a local computer with gmb installed and -lgmb
library introduced to the compiler).
Gap can calculate to a fraction but cannot show the decimal representation.