What is the rate that people accidentally fly with guns on planes?

There are two interesting numbers that can be put together here.

1) Apparently, past security checks have indicated that TSA fail to detect knives and guns about 70 percent of the time.

2) Also this year "Nationwide, more than 1,100 firearms have been discovered at TSA checkpoints from January through September."

If you put those two numbers together, you get the implication that more than 3,422 firearms improperly made it on to flights in the US this year (=1,110 * (70/30) * (4/3)).  (Likely this is lower because the failure to detect rate is probably lower for guns and higher for knives (which are thinner and easier to hide).)  That seems like a lot of guns on planes, yet how many problems arose from all these guns?  It looks to me like there were zero problems.  Obviously, a lot of these people don't know that they have a gun with them when they are flying, so it provides only limited information on how safe it is for people to knowingly fly with guns on planes.

I started thinking about this issue because of this story in the Detroit News: 

Anthony Adams, formerly a Detroit deputy major and school board president, was arrested Thursday afternoon at Detroit Metro Airport after a loaded gun was found in his carry-on bag at a security checkpoint.

Michael Conway, a spokesman for the airport, confirmed Adams was arrested and his .40 caliber handgun confiscated Thursday around 5 p.m. as he was going through McNamara Terminal airport security.

"He was very apologetic. He said he forgot it was in the bag. He and his wife were rushing to get on the plane," said Conway. Adams served as deputy mayor under Kwame Kilpatrick and is among the defense witnesses in the bid-rigging trial of Bobby Ferguson. Adams served as the president of the school board for Detroit Public Schools until his resignation last year.

Conway said police took Adams for questioning and determined the gun was registered to him and he had a permit to carry it. They released him. He missed his flight but was able to fly later, he said. . . .