Muslims believe in Atonement

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I was reviewing the Fifth Sura from the Qur'an, and thinking about what you wrote Seattle, namely that Jews and Muslims have no need for Atonement, knowing that Jews believed in the Yom Kippur, I also remembered the following.

A few verses from the fifth Sura al-Maida may help to shed light on the issue of Islamic interpretation of atonement: (Sura al-Maida 5: 45) Eye for eye and whose waives his right shall find atonement. (Sura al-Maida 5:13) Lend Allah a goodly loan and I will expiate your bad deeds and admit you to Paradise. (Sura al-Maida 5:89) Allah will not take you task for your empty oaths, but will do so for oaths taken in earnest. The expiation for that is to feed ten poor folk or free a slave. Whoever lacks means, let him fast three days. That is expiation for your sworn oaths.

We ordained therein for them: "Life for life, eye for eye, nose or nose, ear for ear, tooth for tooth, and wounds equal for equal." But if any one remits the retaliation by way of charity, it is an act of atonement for himself. And if any fail to judge by (the light of) what Allah hath revealed, they are (No better than) wrong-doers. Sura al-Maida 5:45 (Yusuf Ali's translation)
...And Allah said: "I am with you: if ye (but) establish regular prayers, practise regular charity, believe in my messengers, honour and assist them, and loan to Allah a beautiful loan, verily I will wipe out from you your evils, and admit you to gardens with rivers flowing beneath; but if any of you, after this, resisteth faith, he hath truly wandered from the path or rectitude." Sura al-Maida 5:12 (Yusuf Ali's translation)
Allah will not call you to account for what is futile in your oaths, but He will call you to account for your deliberate oaths: for expiation, feed ten indigent persons, on a scale of the average for the food of your families; or clothe them; or give a slave his freedom. If that is beyond your means, fast for three days. That is the expiation for the oaths ye have sworn. But keep to your oaths. Thus doth Allah make clear to you His signs, that ye may be grateful. Sura al-Maida 5:89 (Yusuf Ali's translation)

The tentative inference from these texts is that almsgiving, the release of slaves and fasting are means of expiation. Muslims can never be sure whether such expiation covers particular sins, or whether expiation entitles them to enter Paradise. Perhaps Muhammad himself was in doubt as regards the efficacy of such casual expiation.

Kiffaratudh-dhunub, atonement for sins, figures as a technical term in some writings. This consists of the four duties, prayer, fasting, alms and pilgrimage. Even the simple Confession, "There is no God but Allah and Muhammad is his Apostle," merits a measure of pardon according to the Hadith. What transpires from the foregoing is that atonement for Muslims has to do with good works, ritual or ethical and the balancing of existential accounts: The weighing on that day will be just. Those whose scales are heavy will prosper, those whose scales are light will suffer loss (Sura al Araf 7:8)

The balance that day will be true (to nicety): those whose scale (of good) will be heavy, will prosper: Sura al Araf 7:8 (Yusuf Ali's translation)

Expiation has its near synonym, fidyah or ransom (or substitute). A sick man, who misses out on some of his fast days pays a ransom by feeding the poor. A similar ransom is demanded from those who default on pilgrimage (Sura al-Baqarah 2:196). But no ransom avails for hypocrites and infidels on the great accounting day (Sura al-Hadid57:15).

And complete the Hajj or 'umra in the service of Allah. But if ye are prevented (From completing it), send an offering for sacrifice, such as ye may find, and do not shave your heads until the offering reaches the place of sacrifice. And if any of you is ill, or has an ailment in his scalp, (Necessitating shaving), (He should) in compensation either fast, or feed the poor, or offer sacrifice; and when ye are in peaceful conditions (again), if any one wishes to continue the 'umra on to the hajj, He must make an offering, such as he can afford, but if he cannot afford it, He should fast three days during the hajj and seven days on his return, Making ten days in all. This is for those whose household is not in (the precincts of) the Sacred Mosque. And fear Allah, and know that Allah Is strict in punishment. Sura al-Baqarah 2:196 (Yusuf Ali's translation)
"This Day shall no ransom be accepted of you, nor of those who rejected Allah." Your abode is the Fire: that is the proper place to claim you: and an evil refuge it is!" Sura al-Hadid57:15 (Yusuf Ali's translation)

Man provides his own ransom to make satisfaction for his own misdeeds, and then he hopes for the best. Man's destiny has become a matter of commercial transaction; man bargaining with Allah, as one might bargaining with Allah in an Arabian market, the Souk. Until the day reckoning, a sinner cannot tell whether his balance is sufficient to balance his sins. Isaiah evaluated human actions more realistically: Our works of righteousness are as filthy rags (Old Testament Isaiah 64:6). Having been made aware of his uncleanness, he had cried out, Woe is me. I am undone.

All of us have become like one who is unclean, and all our righteous acts are like filthy rags; we all shrivel up like a leaf, and like the wind our sins sweep us away. Isaiah 64:6 (NIV)
"Woe to me!" I cried. "I am ruined! For I am a man of unclean lips, and I live among a people of unclean lips, and my eyes have seen the King, the LORD Almighty." Isaiah 6:5 (NIV)

This mercantile frame of mind leads to arithmetic ethical calculations. For example, Ibn abi Laila and a Syrian were traveling along a road when a porter passed by selling pomegranates. The Syrian took one and hid it in his sleeve. Abi Laila couldn't believe his eyes. A poor beggar passed them, and Syrian gave him the fruit he had stolen. The Syrian replied that the theft was one bad deed, while the good act of giving one would get the reward of ten. As a result, thief would have a greater reward than the non-thief.
Another authority, Khashnam, assures us that one fornication is not as bad as 10,000 lies. This stress on the letter of the law led a Bedouin thief, who pulled goods out of a tent with along crooked camel stick, to explain, it was not I that stole but the stick.