Answer:
C. 0.4653
Step-by-step explanation:
This a conditional probability problem where Bayes theorem should be applied.
Applying Bayes theorem,
\(P(\text{No heart attack}\ |\ \text{Correctly tested})=\)
\(\dfrac{P(\text{Correctly tested}\ |\ \text{No heart attack})\cdot P(\text{No heart attack})}{P(\text{Correctly tested})}\)
\(P(\text{Correctly\ tested}\ |\ \text{No\ heart\ attack})=67\%=0.67\)
\(P(\text{No\ heart\ attack})=1-P(\text{heart\ attack})=1-0.7=0.3\)
\(P(\text{Correctly\ tested})=[P(\text{No\ heart\ attack})\times P(\text{Correctly\ tested})]+[P(\text{Heart\ attack})\times (\text{Incorrectly\ tested})]\)
\(=[0.3\times 0.67]+[0.7\times 0.33]=0.432\)
Putting the values,
\(P(\text{No\ heart\ attack}\ |\ \text{Correctly\ tested})=\dfrac{0.67\times 0.3}{0.432} =0.4653\)
There is a probability of 0.4653 or 46.53% chance that she will not have a heart attack even though the test predicts that she will.
