One more thing. Vito and Michael knew very well that Barzini cannot propose "genuine" peace talk.
His goal is to take over the Corleone family, and the best decision for him is to get rid of Mike. Even if they succeeded carefully decieving him about their knowlege of his implication in Sonny's death and attempts at Vito and Michael, they knew both that there's a place on earth for one of them only, Either Barzini or Mike. The winner will be the first to hit. So Mike expected B. to act quickly, and in the most safe way - as Mike says in the book, "Barzini will set me up through somebody close that, supposedly, I can't suspect."
The rest is pure logic- and Mike was always good at logic...