Their vs. they're vs. there

Look at the following sentences. The sentences contain the words their, they're or there. Try to find a pattern and see if you can discover when you should use their, there or when you should use they're.
 

The blue one is their house.
Just ask them what their plans are.
Their mother is very beautiful.

 

They're not very happy today.
They're a fun group.
Do you know they're going to the beach.

 

That is my house over there.
There is Sarah!
Why were you over there in the first place?