The answer is "yes," at least if the number of defective marbles is sufficiently small. Surprisingly, the procedure involves a strong dose of randomness. It's a nice example of a new and very active research area called "compressed sensing" (CS), that spans mathematics, signal processing, statistics, and computer science, and has many surprising applications. In this talk I'll explain the central ideas, which require nothing more than straightforward linear algebra and a bit of probability. I'll then show some applications, including how one can use this to build a high-resolution one-pixel camera!