Why will a person who is born into a kaafir family and dies as a kaafir be punished?
Question

                We as Muslims, beleive in destiny and fate, so if it was written in the fate of an infant to be born in a non-beleivers family, Why would that indiviual go to hell when he grows up and dies ?.
Praise be to Allah.Yes, we believe in al-qadaa’ wa’l-qadar (divine decree), but first and foremost we believe in the perfection and justice of Allaah, and that He is far above being unjust towards His slaves in the slightest. Allaah says (interpretation of the meaning): 
“Truly, Allaah wrongs not mankind in aught; but mankind wrong themselves”
[Yoonus 10:44]. 
And there are many similar, well known verses.  
Above all, we know that Allaah is Kind and loves kindness in all things; He is Most Gracious, Most Merciful, the Bestower of vast mercy that is greater than His wrath and vengeance, as it says in the hadeeth of Abu Hurayrah (may Allaah be pleased with him) according to which the Prophet (peace and blessings of Allaah be upon him) said: “Allaah wrote a decree before He created the universe (saying); My mercy prevails over My wrath, and it is written with Him above the Throne.” Narrated by al-Bukhaari (7554). 
Because of that, He has to allow His slaves a respite and be patient with them. The Prophet (peace and blessings of Allaah be upon him) said: “… there is no one to whom warnings are more than Allaah, and because of that He sent (the Prophets) with glad tidings and warnings.” Narrated by al-Bukhaari (7416). 
Once this wise principle is established in your mind, without which no one’s faith is sound, then you may refer to it for every matter of confusion that is caused by ignorance or specious arguments, and you will find that your mind will be quickly put at ease. You know that Allaah is more merciful towards you and all His slaves than a mother to her child. It was narrated from ‘Umar ibn al-Khattaab (may Allaah be pleased with him) that some prisoners were brought to the Messenger of Allaah (peace and blessings of Allaah be upon him), and there was a woman among the prisoners who was searching for someone. When she found a small boy among the prisoners, she clasped him to her and started to breastfeed him. The Messenger of Allaah (peace and blessings of Allaah be upon him) said to us: “Do you think that this woman would throw her child into the fire?” We said: No, by Allaah, she would never do that if she is able not to. The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Allaah is more merciful towards His slaves than this woman is towards her child.” 
Narrated by al-Bukhaari (5999) and Muslim (2754). Allaah, the Most Gracious, Most Merciful, is more merciful towards that person who you are asking about and arguing about than you or he himself. If he deserves the mercy of Allaah, then the mercy of Allaah will undoubtedly encompass him, but if he is one of the stubborn then you had better focus on what is good for you and what may save you. 
… “so be not a pleader for the treacherous” [al-Nisa’ 4:105]. 
Beware of neglecting this principle even for a moment, then beware of arguing about the divine decree before you understand it. You should understand that the divine decree is like the sun; when you believe in it and surrender to it, on the basis of Qur’aan and Sunnah, then your mind will be at ease, like the one who walks in the daylight. 
But if you examine it too closely and look too deeply, what does the one who stares at the disk of the sun gain except damage to his eyes? 
If we look at what you have asked about, so that you will not think that what we have said above is a way to avoid the question, you should note that Allaah has given every man the potential to distinguish good from bad (interpretation of the meaning). Allaah says (interpretation of the meaning):                                        
“Have We not made for him two eyes?”
[al-Balad 90:8] i.e., two eyes with which to see; 
“And a tongue and two lips”
[al-Balad 90:9], i.e., a tongue with which to speak that which is in his heart, and lips with which to speak. 
“And shown him the two ways”
[al-Balad 90:10] – Ibn Mas’ood said: i.e., good and evil. 
And Allaah says (interpretation of the meaning): 
“Verily, We have created man from Nutfah (drops) of mixed semen (sexual discharge of man and woman), in order to try him, so We made him hearer and seer.
3. Verily, We showed him the way, whether he be grateful or ungrateful”
[Al-Insaan 76:2-3]
From the above verses it is clear that Allaah has given man the means and tools to find out the ways of good and evil. 
Moreover, Allaah has not given these tools only in order to judge people against them; rather He sent them warnings and gave them clear guidance in the Revelation that came to them from heaven. Allaah says (interpretation of the meaning): 
“Messengers as bearers of good news as well as of warning in order that mankind should have no plea against Allaah after the (coming of) Messengers”
[al-Nisa’ 4:165] 
“And We never punish until We have sent a Messenger (to give warning)”
[al-Isra’ 17:15]. 
Because of that, Allaah rebukes the kuffaar for not heeding the warnings and guidance that Allaah sent to them, and for ignoring His Messengers and proof. He says (interpretation of the meaning): 
“’O you assembly of jinn and Mankind! Did not there come to you Messengers from amongst you, reciting unto you My Verses and warning you of the Meeting of this Day of yours’  They will say: ‘We bear witness against ourselves’  It was the life of this world that deceived them. And they will bear witness against themselves that they were disbelievers.
131. This is because your Lord would not destroy the (populations of) towns for their wrongdoing (i.e. associating others in worship along with Allaah) while their people were unaware (so the Messengers were sent)”
[al-An’aam 6:130-131]
As for the one who followed his father in kufr, and died a kaafir, Ibn al-Qayyim (may Allaah have mercy on him) said: 
A distinction is made between the one who imitated his forefathers even though he know the truth, and one who imitated his forefathers and had no way of knowing the truth. The two categories do exist, so the one who knew the truth and turned away from it is negligent and has no excuse before Allaah. As for the ones who were unable to ask and learn, and had no way of knowing the truth, they also fall into two categories: 
1-Those who wanted guidance and sought it but were unable to find it because there was no one to guide them. They come under the same ruling as those who lived in the intervals between Prophets and those whom the call did not reach.
2-Those who did not care and did not seek the truth and it never occurred to them to follow a way other than that which they were following.
The former says: O Lord, if I knew a religion that is better than that which I am following, I would follow it and leave that which I am following; but the latter is content with that which he is following, and would not prefer anything else or seek any other way. It makes no difference whether they were able to do that or not, as both of them are incapable. The former are like those who sought the true religion during the intervals between Prophets but did not find it, so they turned away from it after exhaustive efforts to find out, out of inability and ignorance; whereas the latter are like those who never sought it, rather they died in shirk. End quote. Tareeq al-Hijratayn, 678.
 Even if he sought it he would be unable to find it, and there is a difference between the inability of the seeker and the inability of the one who does not care. Think about this point. Allaah will judge between His slaves on the Day of Resurrection by His wisdom and justice, and He will not punish anyone but those against whom proof is established by means of the Messengers. This case is clear and definite. As for whether proof is established against a specific person, or not, this is something concerning which no one can intervene between Allaah and His slaves. This comes under the rulings on reward and punishment. As for the rulings in this world, it is to be dealt with as it appears to be; the children and insane of the kuffaar are kaafirs according to worldly rulings, and they come under the same rulings as their guardian. 
With regard to those whom the call of Islam did not reach, and against whom no proof was established by means of the Messengers, according to the more correct view, they will be tested on the Day of Resurrection. A Messenger will be sent to them there, and those who obey him will enter Paradise and those who disobey him will enter Hell, as it says in Musnad al-Imam Ahmad (18566) and elsewhere from the hadeeth of al-Aswad ibn Zaree’ that the Prophet of Allaah (peace and blessings of Allaah be upon him) said: “There will be four on the Day of Resurrection: a deaf man who could not hear anything, an imbecile, a senile old man and a man who died during the interval between two Prophets. As for the deaf man, he will say: ‘Lord, Islam came but I did not hear anything.’ As for the imbecile, he will say: ‘Islam came but the children were throwing dung at me.’ As for the senile old man, he will say, ‘Lord, Islam came but I did not understand anything.’ As for the one who died during the interval between two Prophets, he will say: ‘Lord, no Messenger of Yours came to me.’ They will give their pledge to obey Him, then a Messenger will be sent to tell them to enter the fire.” He said:  “By the One in Whose hand is the soul of Muhammad, if they enter it, it will be cool and safe for them.” According to another report: “The one who enters it, it will be cool and safe for him, and the one who does not enter it will be dragged to it.” Narrated by Ahmad. 
And Allaah is the source of strength.