Answer-
The probability that Erin will not have a heart attack and the test predicts that she will is, 46.5% .
Hint- This a conditional probability problem where Bayes theorem should be applied.
Solution-
Applying Bayes theorem,
\(P(No\ heart\ attack\ |\ Correctly\ tested)\)
\(=\frac{P(Correctly\ tested\ |\ No\ heart\ attack)P(No\ heart\ attack)}{P(Correctly\ tested)}\)
\(P(Correctly\ tested\ |\ No\ heart\ attack)=67\%=0.67\)
\(P(No\ heart\ attack)=1-P(heart\ attack)=1-0.7=0.3\)
\(P(Correctly\ tested)=[P(No\ heart\ attack)\times P(Correctly\ tested)]+[P(Heart\ attack)\times (Incorrectly\ tested)]\)
\(=[0.3\times 0.67]+[0.7\times 0.33]=0.432\)
Putting the values,
\(P(No\ heart\ attack\ |\ Correctly\ tested)=\frac{0.67\times 0.3}{0.432} =0.465\)
∴ There is a 46.5% chance that Erin will not have a heart attack even though the test predicts that she will.
