Solution for : https://www.cut-the-knot.org/Probability/GettingFromAtoB.shtml

getA2B_solution.py

from sympy import binomial as b
def getA2B(X,Y,x,y):
  U,D,s = X==x,Y==y,x+y
  u = sum([b(s,i) for i in range(0,y)])
  d = sum([b(s,i) for i in range(y+1,s+1)])
  r = u*U+b(s,y)+D*d
  return r/2**s
print('B(3,4) C(3,3) => ',getA2B(3,4,3,3))
# Problem represnetation :
# Enter destination coordinates B(X,Y) and intercept point C(x,y)
# By definition input should obey X >=x and Y>=y
# Example:
#                             B
#(0,4) -- (1,4) -- (2,4) -- (3,4)
#  |        |        |        |
#(0,3) -- (1,3) -- (2,3) -- (3,3) → C
#  |        |        |        |
#(0,2) -- (1,2) -- (2,2) -- (3,2)
#  |        |        |        |
#(0,1) -- (1,1) -- (2,1) -- (3,1)
#  |        |        |        |
#(0,0) -- (1,0) -- (2,0) -- (3,0)
#  A