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Marshall comments on The Least Convenient Possible World - Less Wrong
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I'm not sure if I'm evading the spirit of the post, but it seems to me that the answer to the opening problem is this:
If you were willing to kill this man to save these ten others, then you should long ago have simply had all ten patients agree to a 1/10 game of Russian Roulette, with the proviso that the nine winners get the organs of the one loser.
Throwing a die is a way of avoiding bias in choosing a person to kill. If you choose a person to kill personally, you run a risk of doing in in an unfair fashion, and thus being guilty in making an unfair choice. People value fairness. Using dice frees you of this responsibility, unless there is a predictably better option. You are alleviating additional technical moral issues involved in killing a person. This issue is separate from deciding whether to kill a person at all, although the reduction in moral cost of killing a person achieved by using the fair roulette technology may figure in the original decision.
But as a doctor, probably you will have to choose non-randomly, if you want to stand by your utilitarian viewpoint, since killing different people might have different probabilities of success. Assuming the lest convenient possible world hypothesis, you can't make your own life easier by assuming each one's sacrifice is as likely to go well. So in the end you will have to assume that one patients sacrifice will be the "best", and will have to decide if you kill them, thus reverting to the original problem.