P(correct guess) =
P(we were shown the higher number H)
* P(we guessed "high" given H)
+ P(we were shown the lower number L)
* P(we guessed "low" given L)
=> (plugging in from the original selection method)
P(correct guess) = (1/2) * (1 - F(H)) + (1/2) * F(L)
=>
P(correct guess) = (1/2) (1 - (F(H) - F(L)))
=> Since F(H) - F(L) > 0 (by assumption)
P(correct guess) = (1/2) (1 - (a positive value))
=>
Therefore,
P(correct guess) < 1/2.