A Counterintuitive Probability Game
The results Cover claims hold. When the range of C, the third random number, overlaps that of A and B, the win rate is over 50%. When the third random number, C, falls in is the same possible range as that of the other two numbers, 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 the entire Real number line as the possible range for C is to ensure that you at least sometimes choose a value in between the ranges the other player 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.