pick any everywhere non-zero continuous prob density function f(x) (for eg, the guassian N(0,1) )

The (randomized) algorithm is:

1.pick a hand (left or right) randomly. Let's say the number you see is a.

2.With prob \int_{a}^{+\infinity} f(x)dx, we guess the other number is larger

  With prob \int_{-\infinity}^{a} f(x)dx, we guess the other number is smaller

             (\int_a^b means we take intergal from a to b)

Analysis:

w.l.o.g  a(in left hand) > b(in right hand)

Prob(a correct guess)= 1/2 (you saw a) * \int_{-\infinity}^{a} f(x)dx (you guess b<a)

                      +1/2 (you saw b) * \int_{b}^{+\infinity} f(x)dx (you guess a>b)

                    = 1/2(1+ \int_{b}^{a} f(x)dx ) >1/2