The Auctioneer's Empty Crate

At a charity auction, three identical sealed crates sit on stage: one holds a vintage motorcycle, the other two hold hay bales. You commit to crate 1. The auctioneer — who knows exactly where the motorcycle is and always opens a hay-bale crate you did not choose — pries open crate 3 and shows hay. He then offers to let you trade crate 1 for crate 2. Should you switch, and what are the win probabilities for staying versus switching?