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
TARKIN
The princess! Put all sections on alert!
VADER
Obi-Wan is here. The Force is with him.
TARKIN
If you're right, he must not be allowed to escape.
VADER
Escape is not his plan. I must face him alone.
Of course,
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.
TARKIN
Are they away?
VADER
They have just made the jump into hyperspace.
TARKIN
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.
HAN
Not a bad bit of rescuing, huh? You know, sometimes I even amaze myself.
LEIA
That doesn't sound too hard. Besides, they let us go. It's the only explanation for the ease of our escape.
HAN
Easy... you call that easy?
LEIA
Their tracking us!
HAN
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?