Can You Solve the Prisoner Boxes Riddle?

Here's a riddle. Ten band members have had their musical instruments placed randomly in boxes. Each band member gets five shots at opening boxes, trying to find their own instrument. (Thus, a 50% chance of each individual finding the desired instrument.) They're not allowed to communicate about what they find. If the entire band fails to find their instruments, they're all fired...and the odds of them all finding their instruments via random guessing is 1 in 1,024. But the drummer has an idea that will radically increase their odds of success. What's the big idea?

This puzzle is based on the 100 Prisoners Problem, though it has been simplified just a bit, and lightened up to be about a band (in the original 100 Prisoners Problem scenario, the punishment is death rather than loss of a job). Here's what the TED-Ed video gives us as the rules:

1. Instruments have been randomly placed in 10 boxes. 2. The pictures on the boxes don't necessarily correspond to the instruments inside. 3. Each musician can open up to 5 boxes. They have to close all of the boxes they open. 4. All 10 musicians must find their own instruments. 5. The musicians can't in any way communicate to each other what they find.

Given all this, it seems rather hopeless. But think on it. What could you do if you talked it through beforehand, as this puzzle allows? Watch this video for the setup, and then pause it when it tells you to, in order to have a good long think. I'd like to think I could come up with a solution like this, were my band (or band of prisoners) in a similar situation. Enjoy:

For more on the original 100 Prisoners Problem, check out this nice writeup. There's also more info from TED-Ed (check the "Dig Deeper" bit for useful links).