When were Kay and the children in danger ?
After the Tahoe hit, of course.
This awnser seems easy to me. Mike says to Frankie concerning the Tahoe hit...
"In my home....in my bedroom, where my WIFE sleeps....where my CHILDREN come to play w/ their toys."If someone shoots up your bedroom it's time to grab the revolver. Mike leaves Tom in charge, saying...
"I'm trusting you with the lives of my WIFE and my CHILDREN.He then leaves to meet w/ Roth and Pentangeli(the potential assassins) to put them in check, and delay any further attempts they may have in mind. After all, if the assassins will shoot up his bedroom and cause him to shield his wife from bullets, who's to say they won't try something more drastic....
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As for Micheal's quest for legitimacyAfter Kay calls him out for his five year legitimacy-plan NOT working, he say's....
"I know, I'm trying darling," I can't see any reason to think that Mike did not think the gambling plans w/ Roth would lead to legitimacy. He thought that the Corleone monopoly on legalized gambling would in fact lead to long last legitimacy. I believe that as Mike dances with Kay at the party he believes Johnny's message about Roth going along w/ the Klingman 'move out' to be true. He thinks that as soon as the move to legalized gambling goes through the Corleones will be legit....But....the next scene in the movie proves him wrong. It's all a dream, b/c Roth sends gunmen to kill Mike. Mike's dreams of legitimacy are just that, a dream. The quest will go on, and on....