I read an interesting math paper by Thomas Cover that I struggled to believe at first. I recommend you read it before continuing reading this post (it’s short). Thus, I decided to test out the claim using Python.
The results hold. When the range that the third random number, C, falls in is the same as the range which the other two numbers may be selected from, the win rate is 66%. When C is a fixed number equally between the upper and lower limit of what the other two numbers may be, the win rate is 75%.
Thus, the benefit of using a wide range of C on the Real number line is to ensure that you are at least sometimes choosing a value in between the ranges in which the other play is selecting numbers from (they don’t have to choose anywhere around 0). If your C is never between their two numbers, your probability of winning is indeed 1/2.