Carey survived an appeal for caught behind when on 72 at the Adelaide Oval yesterday, playing at a ball from Josh Tongue that went through to the wicketkeeper.
England reviewed the decision and there was a noise, but the spike on the review technology did not line up close enough to when the ball passed the bat.
The founder of BBG, who operate Snicko, has since admitted human error was at fault and Carey must have hit the ball.
Carey went on to make 106, his maiden Ashes century, as Australia reached 8-326 at stumps on day one of the third test. They were 6-245 at the time of the flashpoint.
After play, the wicketkeeper-batter conceded he had heard the noise as the ball went past the bat.
England then spoke with match referee Jeff Crowe, who has since agreed that a fault occurred and handed the review back to England.
It meant the tourists started play today with two reviews left, rather than one.
The England and Wales Cricket Board are expected to raise the issue with the International Cricket Council around ensuring the reliability of the technology, with Cricket Australia also asking questions of the operators.
"Given that Alex Carey admitted he had hit the ball in question, the only conclusion that can be drawn from this, is that the Snicko operator at the time must have selected the incorrect stump mic for audio processing," BBG founder Warren Brennan said in a statement last night.
"In light of this, BBG Sports takes full responsibility for the error."
Carey himself admitted on Thursday morning he had been relieved when he saw the gap between the spike on Snicko and the ball passing his bat.
"You need a bit of luck in cricket sometimes and yesterday I had that luck," Carey said on SEN.
"I was just happy it was out of sync. Once you see the umpire doesn't give it and the third umpire doesn't give it you move on pretty quickly.
"After play we heard about what goes with that. But after that it was nothing out of the ordinary, stay in the moment, stay focused and keep on batting."
England bowling coach David Saker made no secret of his frustration after play on day one.
"They boys were pretty confident. I think the calibration of Snicko (has been) out quite a bit and that's been probably the case for the series.
"There's been some things that don't really measure up.
"And at that stage, I can't remember the exact score, but it was a pretty important decision. So those things hurt but we'll get through it.
"But you'd think in this day and age, you would think the technology is good enough to pick things up like that."
Cricket Australia boss Todd Greenberg told local radio station SEN the governing body was seeking answers, Reuters reports.
"It certainly caused me some heartburn because the whole idea of technology is to take away the clanger or the howler,” Greenberg said.
“From what I can understand having dug into it last night and this morning is human error. There’s two human errors there - one is the actual decision from the umpire and then there’s supposed to be a fail-safe with the technology and it didn’t happen.
“In my view it’s not good enough. We’re asking a lot of questions of the provider through the broadcasters and we’re hoping to get some answers so we can be assured it won’t happen again.”
The contentious incident came two years after Carey was cast as the villain of the 2023 Ashes when he threw down Jonny Bairstow's stumps in the second test at Lord's.
The run-out was legal but triggered a "spirit of cricket" debate, and enraged England fans jeered Carey for the rest of the drawn series.
Yesterday's escape may do little to boost his popularity in England but in Adelaide Carey had the home crowd chanting his name and he savoured a standing ovation when he reached his ton.
"You're not always going to be liked. Sometimes it depends which side of the fence you're on," said Carey.
"There's always heroes and villains."
Australia were dismissed for 371 in the first innings early on day two in Adelaide.
Australia hold the Ashes and lead the series 2-0 after eight-wicket wins in Perth and Brisbane.
- Additional reporting by Reuters
