What is the life expectency for a person with stage 4 throat cancer who is refusing treatment?

Not good : Stage 4 any cancer is not good, some better than others. With head and neck cancers though unlike some others, cure is possible with aggressive mutimodality treatment. Usually without any treatment the tumor grows locally, cause local symptoms and metastasize to lymph nodes and distant organs. Consider at least palliative treatment, it can improve quality of life for some time. 6-24 months range.