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%. The test predicted that Erin will not have a heart attack. What is the probability after the test was taken that she will have a heart attack?
0.4051
0.5010
0.4653
0.6632

Answer 1

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.