Would your answer change if you were somehow legally obligated to those 200 people? Say you were a doctor who was in charge of those people?

My thanks to dhruv on a most excellent question!

That does APPEAR to pose a problem, doesn't it?  However, then the question is a Hobson's Choice: you clearly have a legal obligation to your own family as well!

Let me pose you a different question: would you save one of your children over another given that if you do not choose, you lose them both?

Can you be found morally wrong for choosing one life over another when you cannot save both?  I actually do not think that you can.

Such was the issue in the movie, Sophie's Choice, and her choice haunted her for the rest of her life.  However, the choice to give up the daughter rather than the son to the Nazis was not the real dilemma.  It was her willingness to engage with the vile Nazi in the first instance.  By attempting to curry favor with evil, she caused the loss of her daughter.

Still, what you ask is an excellent question.  I am not a believer that one should merely do things because the law demands it - after all, the law itself could be immoral.  If I strictly chose to follow the law in all matters, that would have made me a deontologist when it comes to ethics.  Instead, I evaluate things through three different lenses and ask three different questions:

1. What does the law require of me?  In the case of your question, the law requires that I save both but I cannot save both.  If I help the 200 random strangers, I have actually committed murder of my child.  If I help my child, I have actually murdered 200 random strangers.  While one has more people than the other, the piling on of murders is, strictly speaking, not a legal question but rather a question of consequences, which I will detail in the next question I would ask. As such, the law's requirements are immaterial. DEONTOLOGICAL ETHICS DOES NOT HELP ME HERE.  Note, however, I would still argue that under natural law, I owe a greater allegiance to my own kin than to any stranger, so deontological ethics actually STILL suggests that the very question itself is wrong - I cannot owe a greater legal responsibility to others than to my own flesh and blood but, for the sake of argument, I will allow us to consider this equal balancing of responsibilities and see how it plays out.

2. What about the outcome of it all? Pure utilitarian ethics suggests that I ought to save the 200 random individuals, I am obviously not a pure utilitarian. Still, the problem is that I now have a legal obligation to both groups.  Having earlier rejected the saving of 200 random strangers, we now must weight carefully this decision.  From that utilitarian calculus, it would strongly suggest that we ought to value 200 lives over 1.  CONSEQUENTIALIST ETHICS SUGGESTS HELPING THE 200 PEOPLE.

3. Could I respect my decision later?  If I were to choose the random strangers over my own child, I could not respect that decision.  I would consider myself to be a bad person for having done so.  I would not in the least regret the loss of life of 200 to save my 1 child provided I have merely refused to assist as opposed to actively decided to kill.  VIRTUE ETHICS SUGGESTS SAVING MY CHILD.

So now I have the ultimate issue for me: I use a three-legged stool (deontological ethics, consequentialism, and virtue ethics) approach to ethics.  When all three are in alignment, I know that what I am doing is ethical.  When two of the three are in alignment (as in my previous example), I know that it is more likely than not that it is the correct decision.  When I have a split, ethical considerations suggest that neither side is more ethical.  It ceases (for me) to be an ethical question at all. That means that I would still choose my child.  If it is not a question of ethics, then it is not unethical to do so.

Yet, your question does pose an interesting alternative scenario: why I am willing to allow 200 people to die but not to even actively murder even 1?  Let us return to how this differs from when I said that I would not murder another to save my child:

The real point would be whether I am legally obligated for the murder of my child or not if I refused.  I a madman is holding my daughter hostage and demanding that I kill someone else or else he will kill my daughter and assuming that I literally cannot take action against the madman (always the preferable course of action!), then the question is: would I be legally responsible for her murder if refused to carry out the murder of say ONE people to save her?  The answer from a legal perspective is clear: I would not.  Thus in that case deontological ethics says "don't murder", consequentialist ethics is neutral (one life is no different than another - but note that if there is the murder of more than one, the calculus decidedly goes in favor of the strangers), and virtue ethics also says "don't murder" because while I would love to have my daughter in my arms, I would definitely not be able to respect my decision to murder 200 people even to save my own child.

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Finally, let us consider the perverted regime where the law and morality are not in alignment.  This is really the only scenario where I make an "unethical" choice from the perspective of my ethical framework:

You see if my legal obligation is to strangers MORE than my own kin or even EQUAL to my own kin, I would have to say that the "law is an ass" in the case.  I would now have to appeal to natural law over manmade law.  Once again, under natural law, I owe nothing to others except in the special relationship that exists within the family.  As such, I would never owe a duty to 200 strangers that could possibly override my duty to my own family.  That is how I would "solve" this "problem".

From Adam Smith, The Theory of Moral Sentiments:

“Let us suppose that the great empire of China, with all its myriads of inhabitants, was suddenly swallowed up by an earthquake, and let us consider how a man of humanity in Europe, who had no sort of connection with that part of the world, would be affected upon receiving intelligence of this dreadful calamity.

He would, I imagine, first of all, express very strongly his sorrow for the misfortune of that unhappy people, he would make many melancholy reflections upon the precariousness of human life, and the vanity of all the labours of man, which could thus be annihilated in a moment. He would too, perhaps, if he was a man of speculation, enter into many reasonings concerning the effects which this disaster might produce upon the commerce of Europe, and the trade and business of the world in general. And when all this fine philosophy was over, when all these humane sentiments had been once fairly expressed, he would pursue his business or his pleasure, take his repose or his diversion, with the same ease and tranquility, as if no such accident had happened.

The most frivolous disaster which could befall him would occasion a more real disturbance. If he was to lose his little finger to-morrow, he would not sleep to-night; but, provided he never saw them, he will snore with the most profound security over the ruin of a hundred millions of his brethren, and the destruction of that immense multitude seems plainly an object less interesting to him, than this paltry misfortune of his own.

To prevent, therefore, this paltry misfortune to himself, would a man of humanity be willing to sacrifice the lives of a hundred millions of his brethren, provided he had never seen them? Human nature startles with horror at the thought, and the world, in its greatest depravity and corruption, never produced such a villain as could be capable of entertaining it. But what makes this difference? When our passive feelings are almost always so sordid and so selfish, how comes it that our active principles should often be so generous and so noble? When we are always so much more deeply affected by whatever concerns ourselves, than by whatever concerns other men; what is it which prompts the generous, upon all occasions, and the mean upon many, to sacrifice their own interests to the greater interests of others?

It is not the soft power of humanity; it is not that feeble spark of benevolence which Nature has lighted up in the human heart that is thus capable of counteracting the strongest impulses of self-love. It is a stronger power, a more forcible motive, which exerts itself upon such occasions. It is reason, principle, conscience, the inhabitant of the breast, the man within, the great judge and arbiter of our conduct. It is he who, whenever we are about to act so as to affect the happiness of others, calls to us, with a voice capable of astonishing the most presumptuous of our passions, that we are but one of the multitude, in no respect better than any other in it; and that when we prefer ourselves so shamefully and so blindly to others, we become the proper objects of resentment, abhorrence, and execration.

It is from him only that we learn the real littleness of ourselves, and of whatever relates to ourselves, and the natural misrepresentations of self-love can be corrected only by the eye of this impartial spectator. It is he who shows us the propriety of generosity and the deformity of injustice; the propriety of resigning the greatest interests of our own, for the yet greater interests of others, and the deformity of doing the smallest injury to another, in order to obtain the greatest benefit to ourselves.
It is not the love of our neighbour; it is not the love of mankind, which upon many occasions prompts us to the practice of those divine virtues. It is a stronger love, a more powerful affection, which generally takes place upon such occasions; the love of what is honourable and noble, of the grandeur, and dignity, and superiority of our own characters.”