Do we have to probate my mother's house? It is worth $450,000. I am the excurtor and there are no other assets

The need for probate would exist if the real estate were in your mother's sole name at the time of her death. Whether probate is needed for real estate after someone's death is not based on how much it is worth. Perhap the best way to think of whether probate is needed is to ask this question:  If the real estate were being sold, who would be able to sign the deed, and what would give that person the right to sell the real estate that is in somebody else's name?

Yes.  That is the only way to pass title.