Below is the definition of the useless function:

1: function useless(n)

2: if n = 1 then

3: return 1

4: else

5: return useless(random(1, n))

6: end if7: endfunction

where random(1, n) returns a uniformly distributed random integer in the range 1 . . . n.

Assuming an initial call to the function useless(m).

Point: The function will return 1 or will run infinitely.

Q1. How to prove it formally?

Point: Call to random function on line 5 will occurred whenever m is not equal to 1 and it will be infinite until m equals to 1.

Q2. So, how exactly we can calculate the expected no. of calls to random function?

Point: For best case scenario of useless function [useless(1)], calls for random function will be 0.

Q3. But what about the worst case scenario(useless(n) where n not equal to 1), how many times random function will be called?