Fifty Peeks at a Hundred Drawers

A firm's badge room has $100$ numbered drawers holding the $100$ interns' numbered badges in a uniformly random one-per-drawer arrangement. Each intern enters alone and may open at most $50$ drawers seeking their own badge; drawers are shut again before the next intern, and no communication is allowed once searches begin. The cohort graduates only if every intern finds their own badge. Opening $50$ random drawers per intern succeeds with probability $(1/2)^{100}$. Find a strategy that succeeds with probability greater than $30\%$, and compute that probability.