Worried for my adopted dog?

I have recently adopted my deceased father's Jack Russell Terrier.He was the only dog in the house and was the centre of attention.He went everywhere with my Father He is almost 6 and neutered.He has a very fond attatchment to my son. who just loves him, they do everthing together. walks and play and also sleeps with him,sometimes he sleeps with me.But I am concerned about his nature towards me. On a few occasions he has snapped and bitten me, which I might add was completly unprovoked, He was sleeping on Son's bed I went in to see what he was doing, he woke up I spoke to him, I patted him and he just turned on me.He has also done this at other times. But when we are on the couch of a night watching tv he cuddles up to me for a scratch..I feed him and bathe him and look after him in general.Can anyone help me with this problem. If it continues I am afraid I will have to have him put down.I can't take any risks with him,I am a n elderly lady and fear for dog bites. Much appreciated