[sirclueless]

This is why your choice of threshold matters. Your strategy should be to choose a threshold randomly from any distribution that is non-zero everywhere (such as a normal distribution), then choose a hand uniformly at random.

Choosing zero as a threshold deterministically does not work for all choices that Alice can make because, as you say, for some choices such as (10,20) it gives you exactly 50% chance to win. But as long as there is always a non-zero chance that you chose a threshold between Alice's numbers, then there is a > 50% chance that you win the game.