100 prisoners and boxes

100 prisoners numbered 1–100. In a room are 100 closed drawers, each containing one number, forming a random permutation. Each prisoner enters alone, may open up to 50 drawers, then leaves; no communication afterward. They may agree on a strategy beforehand. If every prisoner finds their own number, all go free; otherwise all are executed.

Is there a strategy with success probability much higher than random guessing?