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Hi, the system is special in some sense, I think that you can do the following

var('t')
p=1 - (1 - (1 - (1 - 0.01)^1036)^20)*(1 - t)^(9)
eq1 = t==(2*(1 - 2*p))/((1 - 2*p)*16 + 15*p(1 - (2*p)^1023))
eq1.find_root(-100,100)

and then

p.subs(t=_)