Yes, thanks to Cyan's reply, I've already figured that "it is 11 times more likely to be a diamond when it beeps" than that the beep indicates a false positive. Your reply makes it even more obvious. My whole problem was my ignorance regarding the meaning of the likelihood ratio of testing a random box opposed to the overall probability of a beep. Or in other words, I was unaware that there were actually two different questions being estimated.

Thanks everybody!