Violence has ebbed across Iraq, but insurgents frequently attack security forces and civilians in an attempt to undermine the country's Shiite-led government.
Friday's attacks came as Iraqis marked the first day of a major Muslim holiday, the Eid al-Adha, with traditional prayers. Ammar al-Hakim, a Shiite leader and the head of the Supreme Islamic Iraqi Council, led the prayers in the capital, Baghdad.
The first attack took place in the early morning hours, when gunmen opened fire on a group of policemen at a checkpoint in Buhriz, about 60 kilometers (35 miles) north of Baghdad, police officers said. Two policemen were killed and two were wounded in the shooting in Buhriz, a former insurgent stronghold.
Shortly after sunset, a car bomb exploded near a Shiite mosque in Mahaweel, killing two Shiite worshippers as they were leaving the mosque compound, police officials said. Six worshippers were wounded in the attack. Mahaweel is 56 kilometers (35 miles) south of Baghdad.
Medics in nearby hospitals confirmed the casualties. All officials spoke on condition of anonymity because they were not authorized to talk to the media.