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Casualty Risk is an imprecise metric used in the duty officer assignment system. It reflects the risk your duty officers take of ending up in sick bay following the current assignment. More precisely, it's the chance of being forced to complete the 20-hour, involuntary “Sick Bay” assignment.
It can be found directly beneath the pie chart representing the "chance of success" involved with any particular mission. The player never knows the exact percentage risk of going to sick bay, but instead has only a vague sense of danger, as reflected by words like: "None", "Low", "Medium", "High" and "Extreme". These words are color coded to supply an additional sense of danger, with "None" being a cool blue, "Low" being green, and "Extreme" being red.
Though it sometimes seems as if there's a relationship between the relative success of a mission and whether your officers take a trip to the sick bay, in fact the two components of duty officer play are separate. It is possible to critically succeed a mission but still end up in sick bay. It is also possible to have a disaster, but not end up in sick bay. Still, there is some mathematical influence of the success roll on the casualty risk roll, as you are more likely to go sick bay if your assignment ends in disaster. Because the Casualty Risk metric is not conveyed numerically, however, it's impossible to state with authority what is the relationship between the two.
However, some guidelines can be issued, at least about the extreme ends of the casualty risk scale.
The kinds of missions in which a disaster will not' land your duty officers in sick bay are those with a casualty risk of "None". You're not going to sick bay if you totally blow “Continue Tradition of Captain's Night”, for instance.
The kinds of assignments in which a critical success will send your guys to the doctor are those with a casualty risk of "Extreme". Personal combat missions like “Compete in Unsanctioned Combat Arena” are dangerous enough that you can succeed, but still need a doctor.
Thus, you have to pay attention to the Casualty Risk metric. If you don't want a particular duty officer to be incapacitated for 20 hours, you might consider changing them up, even if it appears they're going to have zero chance of failure. Conversely, you might use duty officers who increase the chance of disaster (but also increase the chance of critical success) if you know that a disaster doesn't carry with it the penalty of sick bay time.