A point mutation on the gene for clotting factor V results in a varient known as 'factor V Leiden'.
It is inherited in an autosomal dominant fashion.
The result of the mutation is that factor V is insensitive to protein C inactivation, resulting in hypercoagulability.
This leads to increased risk of PE/DVTs and in females increased miscarriage.
Merely being factor V leiden positive doesn't necessitate anticoagulation; warfarin etc would only be considered in response to events rather than prophylactically - indeed, up to 7% of all white europeans/north americans have factor V leiden, making it the commonest inherited thrombophilia.
So having looked at a cause of thrombophillia, lets switch to the opposite and consider low platelets...