Question

On discovering that her family had a 70% risk of heart attack, Erin took a treadmill test to check her own potential of having a heart attack. The doctors told her that the reliability of the stress test is 67%. What is the probability that Erin will not have a heart attack and the test predicts that she will?

Answer 1

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.