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You can also use mpmath

from mpmath import * 
mp.dps = 30; mp.pretty = True
f1 = lambda t,p:1 - (1 - (1 - (1 - 0.01)**1036)**20)*(1 - t)**(9)-p  
f2 = lambda t,p:(2*(1 - 2*p))/((1 - 2*p)*16 + 15*p*(1 - (2*p)**1023))-t
findroot([f1,f2],(1,1))
[2.39470409963011871605945052144e-309]
[    0.999398871164408619804930822283]