If I'm really worried about wasting their time, I wait a few days before sending the link. If I haven't gotten a good answer on MO after a few days, it suggests the question isn't an easy one.
]]>In graduate school, you have the convenient thing that if you want to ask Prof. X a question, you can first ask one of X's graduate students. Often that'll get you an answer, but if it doesn't then you know it's a hard question. To some extent MO automates this process, where younger people with more time probably check MO more often, but then the unanswered questions stick around for other people to answer.
]]>From the point of view of the questioner the advantage is that you may get a good answer from an unexpected person. Furthermore, it seems plausible to me that you might get the answer faster on MO as more people will see it. The main disadvantages for the asker are that some people won't speculate as freely in public, and that the askees may not see the question on MO depending on timing (which may result in the question getting answered more slowly or not at all).
From the point of view of the potential answerers the main advantage is that you go to MO when you're in the mood to answer math questions, whereas you might be checking your email for lots of reasons. Furthermore, you're less likely to waste your time answering it, because you can see more easily if it's already been answered.
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