Question

Hunter says she is itchy. There is a test for allergy to cats, but this test is not always right: for people that really do have the allergy, the test says "yes" 80% of the time for people that do not have the allergy, the test says "yes" 10% of the time ("false positive") if 1% of the population have the allergy, and hunter's test says "yes", what are the chances that hunter really has the allergy?

Answer 1

Answer:

first we know the chance of having allergy when test says yes and written P(Allergy|Yes)

Then the formula is

P(Allergy|yes)=$\frac{P(Allergy)P(Yes|Allergy)}{P(yes)}$

Here P(allergy) is probability of allergy=1%

P(Yes|Allergy) is probability of test saying yes for people with allergy=80%

P(Yes) is probability of test saying Yes =??%

we dont know what the general chance of the test saying yes is but we can calculate it by adding up those with and those without the allergy:

1% have the allergy and the test says yes to 80% of them

99% do not have the allergy and the test says Yes to 10% of them

P(Yes)=1%*80%+99%*10%=10.7%

it means that the 10.7% of the population will get a yes result

P(Allergy|Yes)=1%*80%/10.7%=7.48%

P(Allergy|Yes)=about 7%

This is the same result we got on False Positives and False Negatives.