I am modelling both eyes of a person, both of which can be categorized into 3 different states, depending on the visual acuity in each eye. These different states are VA1, VA2 and VA3 (where VA1 is very good vision, VA2 is average vision, and VA3 is bad vision).
So the probabilities for each eye falling into each VA state may be as follows:
Eye 1 VA1 VA2 VA3 0.1 0.2 0.7
Eye 2 VA1 VA2 VA3 0 0.2 0.8
Now my task is to calculate the probability that Eye 1 is the better-seeing eye (BSE). Just glancing at the distributions it is easy to see that Eye 1 will have the higher chance to be the BSE, but I am having trouble quantifying it hehe.
Can anyone help? Preferably I would like to calculate a table which basically says: if Eye 1 'falls' into VA1, then what is the probability that it is the BSE? and if Eye 1 'falls' into VA2, then what is the probability it is the BSE? and finally if Eye 1 'falls' into VA3, then what is the probability that it is the BSE?
Any help would be greatly appreciated. I have tried modelling this problem through different approaches and methods but I keep hitting a brick wall, and I don't have confidence in my results probabilities sometimes get me in a bind like this hehe