Suppose you have two algorithms to predict whether a student can obtain an offer from Harvard University based on some features such as GPA, number of credits, internship, and research experience. The following table shows the ground$-$truth label and the predictions made by the two algorithms. The sensitive attribute considered in this fairness study is gender.

$\backslash$table$[[$Gender$,$Label $($truth$),$Algorithm $1,$Algorithm $2],[$Male$,$Yes,No$,$Yes$],[$Male$,$No$,$Yes,Yes$],[$Male$,$Yes,Yes,No$],[$Male$,$Yes,No$,$No$],[$Female$,$Yes,Yes,Yes$],[$Female$,$No$,$No$,$Yes$],[$Female$,$Yes,No$,$No$],[$Female$,$Yes,Yes,No$]]$

$($a$)$ Determine if the algorithms satisfy demographic parity. Show the derivation.

$($b$)$ Determine if the algorithms satisfy equal opportunity. Show the derivation.

$($c$)$ Determine if the algorithms satisfy equalized odds. Show the derivation.

$15$ points