Page 34 of Judea Pearl's book Causality has a statement that I found hard to prove. Any help would be appreciated:
He says given p(y|x)=.5 for all y,x you can prove that a counterfactual probability Q is equal to the inverse counterfactual Q'.
As an example of this he uses the patient treatment example where y is the boolean value of if a particular patient recovers from a disease and x is the boolean of if they are treated with a particular drug.
Q represents the percentage of people who died after using the drug who wouldn't have died if they hadn't used the drug and Q' is the percentage of people in the nontreatment group who would have died if they had used the drug.
I have been able to reduce it down to a few equations:
$Q = \frac{\sum_i z_i x_i}{\sum_i x_i}$
and
$Q' = \frac{\sum_i (1-z_i)(1-x_i)}{\sum_i{(1-x_i)}}$
where $z_i$ is the boolean statement "person i would have survived if untreated."
I want to show
Q = Q'
Thanks in advance!