I follow Deontological ethics and believe that people have a Negative right from being murdered. I believe this right originates from a negative right from theft. Thus killing without consent (theft/vandalism of life) would be murder. A random person cannot give consent. Therefore choosing to kill the random person would be killing them without their consent and thus murder. Since people have a negative right from being murdered, it would be immoral to choose to kill the random person.
Knowing one of the options is immoral is not sufficient. An immoral choice implies the existence of a moral choice. (Ought implies can)
The other option is letting many people die. (deontological ethics would deem that the word many is irrelevant because the rights of the many are the same as the rights of the one) Here is where we discuss whether letting someone die when you could have delayed that death is immoral. (death is inevitable hence a negative right from being murdered instead of a positive right to life) I believe that it is not immoral to let someone die but it is morally praiseworthy to delay that death. If it were immoral to let someone die when you could have delayed that death then choosing to allocate scarce resources among myriad problems would be a choice without a permissible option. (Ought implies can implies such a scenario cannot exist.)
So the choice as I see it is commit a pair of actions (one morally prohibited, one morally praiseworthy) or do nothing (morally permissible but not morally praiseworthy). Here I believe abstaining from morally prohibited actions is more important that doing morally praiseworthy actions.
Your argument is, if I'm correct, as follows:
1) Killing a random person is morally incorrect.
2) Death is inevitable.
3) Delaying death is morally praiseworthy, but not delaying it is not morally incorrect.
4) Finding a cure for a person is delaying its death.
5) The rights of many are as the rights of one.
6) Finding a cure for many people is morally praiseworthy, killing one random person is morally prohibited. Not finding a cure is not morally incorrect, not killing is morally correct.
7) Answer no.
May I require a further explanation as to point 5?
Also, this reasoning has a major flaw in my opinion, in that it doesn't take into consideration consequences of your action.
Let us think that the random stranger is TRULY a random stranger, not "that one random stranger that will save the world if you don't kill him". What are the consequences of his death? Grief and suffering, possibly negative consequences on a certain number of people. On average, not that many people. What are the consequences of finding a cure for AIDS? Extending the life of millions of people, letting them live a better and more satisfying life, possibly for the better of the world population. Incidentally, screwing with the medic industry and with monopolies.
Now, since your action has consequences, those consequences fall under the morality of your action. This is to avoid the "I didn't kill him, the bullet did" syndrome. Are we capable of understanding the full extension of the consequences? Of course not, we aren't capable of understanding the consequences of ANY action, not to the fullest. But, we can make an estimation. Negating the fact that we can make an estimation and act upon that negates our possibility to understand how to live, therefore questions all of our ethical analyses (unless of course this analyses in based on facts only, without consequences... but see above). Can we make a good estimation of what consequences will have such a vast act? It is, by the law of great numbers, about the same - possibly a bit better - as letting people live without interfering: now, is the direction the people we let live (we may or may not want to consider also WHICH people those are, as AIDS has a specific subset of targets, and some issues may or may not arise depending on our view on some other subjects - i.e. racism, morality of sex with unknown partner etc.) is that direction positive or negative? If the answer is "positive", by letting them live you are acting against the betterment of society as a whole, even if you are acting against a single person. I believe the problem can be reduced to this due to the above analyses. I also believe the answer to be strictly positive, therefore if these are the possibilities I would say the right answer is to accept.
One other thing: is murder ever allowed? This includes war, self-defense, euthanasia, abortion as possible cases.