[tex]P(T\mid D)=0.98[/tex]
[tex]P(T^C\mid D^C)=0.97[/tex]
[tex]P(D)=0.035[/tex]
The law of total probability gives us that
[tex]P(T)=P(T\cap D)+P(T\cap D^C)[/tex]
[tex]P(T)=P(T\mid D)P(D)+P(T\mid D^C)P(D^C)[/tex]
By the same token it tells us that
[tex]P(D)=P(T\mid D)P(D)+P(T^C\mid D)P(D)\implies P(T^C\mid D)=1-P(T\mid D)[/tex]
or more generally that some event [tex]A[/tex] conditioned on another event [tex]B[/tex] is complementary to [tex]A^C[/tex] conditioned on [tex]B[/tex].
So we find that
[tex]P(T)=P(T\mid D)P(D)+(1-P(T^C\mid D^C))(1-P(D))[/tex]
[tex]P(T)=0.98\cdot0.035+(1-0.97)(1-0.035)=0.06325[/tex]
