Program to Find Out the Probability of Having n or Fewer Points in Python

Suppose we are playing a unique game and we have three values n, k, and h. We start from 0 points, then we can select a number randomly between 1 and h (inclusive) and we will get that many points. We stop when we have scored a minimum of k points. We have to find the probability that we have n or fewer points. Here any number can be chosen randomly and the outcomes all have the same probability.

So, if the input is like n = 2, k = 2, h = 10, then the output will be 0.11.

To solve this, we will follow these steps −

• Define a function dp() . This will take path.

  • if path is same as k − 1, thens

    • return (minimum of n − k + 1 and h) / h

  • if path > n, then

    • return 0

  • if path >= k, then

    • return 1

  • return dp(path + 1) − (dp(path + h + 1) − dp(path + 1)) / h

• From the main function, do the following −

• if k is zero, then

  • return 1

• if n < k , then

  • return 0

• return dp(0)

Let us see the following implementation to get better understanding −

Example

 Live Demo

class Solution:
   def solve(self, n, k, h):
      if not k: return 1
         if n < k: return 0
         def dp(path):
            if path == k− 1:
               return min((n− k + 1), h) / h
            if path > n:
               return 0
            if path >= k:
               return 1
            return dp(path + 1)− (dp(path + h + 1)− dp(path + 1)) / h
         return dp(0)
ob = Solution()
print(ob.solve(2,2,10))

Input

2,2,10

Output

0.11