Sree Leela thinks she may be allergic to Bengal gram and takes a test that is known to give the following result:
For people who really do have the allergy, the test says “Yes” 90% of the times.
For people who do not have the allergy, the test says “Yes” 15% of the times.
If 2% of the population has the allergy and SreeLeela’s test says “Yes”, then the chance that chance that SreeLeela does really have the allergy is

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1/11

Cannot be determined from the given data

1/9

6/55

answer is B.

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Detailed Solution

Let A denote the event that the test says “Yes”. Consider two hypothesis;
$H_{0}$ - SreeLeela has the allergy;
$H_{1}$ - SreeLeela does not have the allergy;
Then, $P (H_{0}) = 0.02, P (H_{1}) = 0.98, P (A) (H_{0}) = 0.9, P (A) (H_{1}) = 0.15$
By the Bayes rule

$P (H_{0} | A) = \frac{P (A | H_{0}) P (H_{0})}{P (A | H_{0}) P (H_{0}) + P (A | H_{1}) P (H_{1})} = \frac{0.9.0.02}{0.9.0.02 + 0.15.0.98} = \frac{18}{165} = \frac{6}{55}$

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