Making a bit of small talk, Trooper Patterson found out that Mr. Bailly was a retired Piscataway police officer - the same town where Trooper Patterson grew up. The trooper and his mom later met Bailly and his wife.
"Bailly said that he remembered that street, because he helped deliver a baby there 27 years ago when he was a rookie cop". The call was for a woman in labor. After she had struggled home, her husband phoned a doctor.
Bailly then described the color and style of the house and told Patterson that the baby's name was Michael.
A stunned Patterson extended his hand and told Bailly he was the baby the retired officer delivered on October 5, 1991. "And then he said, 'You know what, I had to deliver a baby on that street when I was fresh on the road, I was fresh out of the academy, '" said Patterson.
The trooper learned of the connection when he pulled over the former officer for "a minor motor vehicle violation", New Jersey State Police said. "Maybe they're close to the odds of a hole-in-one, winning the lottery, or being struck by lightning - but it happened".
Back in 1991, Bailly was a rookie officer and responded to a call about a woman, Karen Patterson, going into labor. But, as they say, it's a small world, after all.
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With help from a doctor on the telephone, Bailly was able to deliver the trooper-to-be.
He said he never issued Bailly a ticket.
"It was awesome", 26-year old trooper Michael Patterson said of the couldn't-make-this-up moment at a press conference on Thursday.
Mr Bailly, who had been on the job for four years, was the first to respond to the incident. She barely made it home.
After Bailly told his story, Patterson reintroduced himself. Bobby Patterson, Trooper Patterson's father, rushed outside, picked up his wife, and carried her inside the house.
Trooper Michael Patterson (left) standing with retired police officer Matthew Baily (right) at a reunion following the traffic stop. Their story has been shared on Facebook more than 7,000 times.