A university that uses normalized capability score for approving or rejecting an applicant. Note

there are two population of applicants 'blue' $($deprived group with protected attribute $A=0)$ and

'red' $($favored group with $A=1).$ Admission is approved, i$.$e$.,y\text{'}=1$ if normalized capability score is

greater than some threshold $t,$ else rejected. $y=1$ indicates that an admitted student graduates

successfully and timely, and $y=0$ indicates admitted student was not successful. The university

wants to admit applicants that will graduate successfully and timely. Distributions are described in

following figure $($not drawn to scale$)$ for both populations. A$.$ Calculate probability P$($y$=1)$ for both $$blue$$ and $$red$$ populations for threshold t$=0\mathrm{.}3$ Is fairness achieved w$.$r$.$t$.$ protected attribute A$?$

B$.$ If threshold t$=0\mathrm{.}35$ is chosen for the blue population, calculate the probability P$($y$=1)$ for the blue population. What threshold value for the red population will ensure demographic parity?