When you consider that this life is a trial and is temporary, whatever of pain and pleasure in it is part of that trial and temporary as well. Many believe that God doesn't exist, yet they still face these problems you speak of. In that context, these problems would be considered as part of what is possible in this life; something that happens that people have to deal with; something out of our hands. Yet when God is put into the picture it takes on a negative connotation and people want to blame Him?
The fact of the matter is that He gave us life and everything we have. Ironically, when we think of life, love, happiness, food, shelter, health and all the many other blessings that God gives us, these same people don't ask the question: Why not worship a god that does all this good? How ungrateful and self-centered.
In answer to your question: A God who does whatever He wills. A God who gives life and can take it away. A God who gave you the very intellect which you use to question His existence. And we should worship Him because he gave us life. We owe him our very lives and the amount of worship we do can never fully repay all the things He does for us. This, even if you don't see the wisdom behind His decrees.
I look forward to your response.
This turns the question into "why should I believe in things I don't like, even if they do exist?", which only leads one to ask the questioner why he himself "believes in medical problems" when he apparently doesn't like them (even if they do exist).
My answer justifiably presupposed that this is not the intended meaning of the question. My answer addresses the question with the intended meaning of the word "good" as it seemed, in my mind, to be defined by the questioner.
If you would like to ask the question "why should I believe in things I don't like, even if they do exist?", then feel free.
Be advised that this a dead giveaway that Tim is about to indulge in his favourite past-time of re-wording things so that they bear little or no relation to what was actually asked.