Mortality and morbidity risks from space radiation exposure are an important concern for astronauts participating in International Space Station (ISS) missions. NASA's radiation limits set a 3% cancer fatality probability as the upper bound of acceptable risk and considers uncertainties in risk predictions using the upper 95% confidence level (CL) of the assessment. In addition to risk limitation, an important question arises as to the likelihood of a causal association between a crew-members' radiation exposure in the past and a diagnosis of cancer. For the first time, we report on predictions of age and sex specific cancer risks, expected years of life-loss for specific diseases, and probability of causation (PC) at different post-mission times for participants in 1-year or multiple ISS missions. Risk projections with uncertainty estimates are within NASA acceptable radiation standards for mission lengths of 1-year or less for likely crew demographics. However, for solar minimum conditions upper 95% CL exceed 3% risk of exposure induced death (REID) by 18 months or 24 months for females and males, respectively. Median PC and upper 95%-confidence intervals are found to exceed 50% for several cancers for participation in two or more ISS missions of 18 months or longer total duration near solar minimum, or for longer ISS missions at other phases of the solar cycle. However, current risk models only consider estimates of quantitative differences between high and low linear energy transfer (LET) radiation. We also make predictions of risk and uncertainties that would result from an increase in tumor lethality for highly ionizing radiation reported in animal studies, and the additional risks from circulatory diseases. These additional concerns could further reduce the maximum duration of ISS missions within acceptable risk levels, and will require new knowledge to properly evaluate.