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Cyan comments on Open Thread: February 2010 - Less Wrong
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The likelihood ratio is Pr(beep | diamond) / Pr(beep | empty) = 0.88/0.08 = 11. I was going to say you ought to read the link for "likelihood ratio", but there's nothing there, so you should try the other wiki.
Also, don't think of running the detector over every box; think of testing one box at random.