How did Washington come to inherit his brother's estate, Mount Vernon, as noted in the biography, Washington: A Life?

Upon their father's death, Lawrence was given a small farm near the Potomac River that he dubbed Mount Vernon after a favorite commander in the navy. Lawrence later married and had a daughter, but he soon began to suffer the effects of tuberculosis. Multiple trips springs failed to cure Lawrence, and he died a young man. Upon his death, Lawrence left Mount Vernon to his wife and child, but should they die, Washington would inherit Mount Vernon. This eventually came to pass and Washington would turn Mount Vernon into a large and thriving plantation.