In Star Wars: A New Hope, Vader first tells Tarkin to not worry about Leia and Obi-Wan escaping, as Obi-Wan has no intentions of escaping. (Though, Obi-Wan's intention of not escaping doesn't mean Leia didn't intend to.)
From the script
The princess! Put all sections on alert!
Obi-Wan is here. The Force is with him.
If you're right, he must not be allowed to escape.
Escape is not his plan. I must face him alone.
Vader cuts down Obi-Wan
Leia and the rest of the crew escape aboard the Millenium Falcon. Vader and Tarkin discuss the fact that their escape was allowed by Vader/Tarkin.
Are they away?
They have just made the jump into hyperspace.
You're sure the homing beacon is secure aboard their ship? I'm taking an awful risk, Vader. This had better work.
Leia is a sharp one, though. She knows what's up.
Not a bad bit of rescuing, huh? You know, sometimes I even amaze myself.
That doesn't sound too hard. Besides, they let us go. It's the only explanation for the ease of our escape.
Easy... you call that easy?
Their tracking us!
Not this ship, sister.
A number of (expendable) Imperial Stormtroopers were lost during the escape, along with some TIE Fighters. Were the military personnel involved aware of Vader/Tarkin's plan to allow the escape (thus, partially explaining the horrible aim of the Stormtroopers and pilots), or did Vader/Tarkin just intentionally not send enough resources to properly give chase?