What is the Biblical Basis for a Good person going to hell simply for not believing? The scenario is as follows:


Believer, average life, many mistakes/sins. Asks forgiveness and tries everyday to be better. Goes to heaven.
Non-believer, better life than the believer, fewer mistakes than the believer, tries everyday to be better, succeds more than the believer, knows he has done some bad things in life, regrets them, but doesn't believe in anything because he never witnessed anything supernatural in life so he has no reasons to. Goes to hell.


This scenario was put to debate by a co-worker and I tried to explain how you cannot be saved by your own deeds, but didn't succed to actually give a good argument, so I'm asking for help.

For all the non-believers this scenario is "not fair" as why would a better person go to hell just for not believing. 

Can anyone give some biblical explanations / paragraphs to explain this. Or maybe the premises are wrong?
Romans 3:23 (NIV) for all have sinned and fall short of the glory of
  God


No one is righteous. Better is not enough. God demands 100% holiness. No one can achieve that. Therefore, we need the Blood of Jesus Christ. Only in Jesus, can we have salvation.


  Romans 5:12 (NIV) Therefore, just as sin
  entered the world through one man, and death through sin, and in this
  way death came to all people, because all sinned—


We are all doomed to go to Hell whether we like it or not, whether we are better than others or not. Jesus died for us so that by believing in his death and resurrection, we may be saved.

For this reason Christians are given the command to preach the Gospel to the whole world.


  Matthew 28:19-20 (NIV)
  19 Therefore go and make disciples of all nations, baptizing them in
  the name of the Father and of the Son and of the Holy Spirit, 20 and
  teaching them to obey everything I have commanded you. And surely I am
  with you always, to the very end of the age.”