In another incident, a tea stall owner Rakesh Kumar was shot dead by a group of armed criminals at Choudhary market under Buddha Colony police station late on Monday night.
The third incident took place when a girl was charred to death at Barh area while she was bursting firecrackers.
Sinha and his sister in-law Babita Devi had received bullet injuries while they were returning their home after shopping on Dhanteras on Sunday evening.
They sustained bullet injuries when two motorcycle-borne criminals tried to snatch a gold chain from Babita, who was the wife of one Priyaranjan, branch manager of Central Bank of India, Madhepura.
While Sinha received three bullets in his body, Babita received one. Both were admitted to Rajeshwar Nursing Home in critical condition. Police had recovered five cartridges out of which four were empty.
Additional superintendent of police (ASP), Patna City, Chandan Kushwaha said Sinha had died on Tuesday. "We have failed to get any breakthrough so far but efforts are on."
Station house officer (SHO) of Buddha Colony police station, Vinod Prasad said the cause of Rakeshfs murder was yet to be ascertained. "A case was lodged against five named persons in this connection. Raids are on to trace them," he added.