According to the table,
[tex]P(C\cap T)=\dfrac8{1000}[/tex]
[tex]P(C^C\cap T)=\dfrac{99}{1000}[/tex]
[tex]P(C\cap T^C)=\dfrac2{1000}[/tex]
[tex]P(C^C\cap T^C)=\dfrac{891}{1000}[/tex]
A. By the law of total probability,
[tex]P(C)=P(C\cap T)+P(C\cap T^C)=\dfrac{10}{1000}=0.01=1\%[/tex]
B. By definition of conditional probability,
[tex]P(T\mid C)=\dfrac{P(C\cap T)}{P(C)}=\dfrac{\frac8{1000}}{\frac{10}{1000}}=\dfrac8{10}=0.8[/tex]
That is, the test has an 80% probability of returning a true positive.
C. As in (B), we have by definition
[tex]P(C\mid T)=\dfrac{P(C\cap T)}{P(T)}[/tex]
but we don't yet know [tex]P(T)[/tex]. By the law of total probability,
[tex]P(T)=P(C\cap T)+P(C^C\cap T)=\dfrac{107}{1000}[/tex]
Then
[tex]P(C\mid T)=\dfrac{\frac8{1000}}{\frac{107}{1000}}=\dfrac8{107}\approx0.075[/tex]
That is, the probability that a patient has cancer given that the test returns a positive result is about 7.5%.
D. Simply put, the events [tex]T\mid C[/tex] and [tex]C\mid T[/tex] are not the same, and the difference is derived directly from the fact that [tex]P(C)\neq P(T)[/tex].
