pt1 = (1 - u) ((p ( p w11 + q w12) )/wbar) ;

w11 = 1 ;

w12 = 1 ;

w22 = 1 - s ;

wbar = p p w11 + 2 p q w12 + q q w22 ;

deltap = pt1 - p ;

p = 1 - q ;

FullSimplify[deltap]

((-1 + q) (q^2 s - u))/(-1 + q^2 s)

Reduce[deltap == 0, q]

(q == 1&& -1 + s≠0) || (u == 0&&s == 0) || (s≠0&& (q == -u^(1/2)/s^(1/2) || q == u^(1/2)/s^(1/2)) && -1 + u≠0)

qhat = u^(1/2)/s^(1/2) ;

u = 1.*10^-7 ;

s = 0.01 ;

FullSimplify[qhat]

0.00316228

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