Start with distribution for employee
Good: P(good)=p
Bad: P(bad)=(1-p)
Have a bad employee detector
P(detected as bad|bad)=q1
P(detected as good|bad)=1-q1
P(detected as good|good)=q2
P(detected as bad|good)=1-q2
Fraction of employees detected as good:
pq2+(1-p)(1-q1)
Fraction of employees detected as bad (and therefore fired)
p(1-q2)+(1-p)q1
Suppose you pay a discount on ability to hire new good employees as d (so probability to hire a new good employee is dp instead of p)
So you benefit as long as
p(1-q2)/((1-p)q1)<dp/(1-dp)
Or
(1-q2)/q1 < d (1-p)/(1-dp)
Plugging in
p=.8, d=.5, q1=q2=q (knowing good and bad is equally hard)
(1-q)/q < .167
Break even is then
q = 1/1.167=.857
So if you start with 80/20 distribution and after a while improve your discrimination to 86/14 then firing benefits you even if you are hiring from a 40/60 pool.