A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability 0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5 % of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the probability that, given a positive result, the person is a sufferer.

0.0455

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0.445

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0.1

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0.091

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Solution

The correct option is A 0.0455
$Let us define the following events : E : ‘ The person is sufferer ’ . F : ‘ The test gives positive result ’ . Given P (E) = 0.005 then P (E^{C}) = 0.995 and P (F | E) = 0.95, P (F ∣ ∣ E^{C}) = 0.1 We have to calculate P (E | F) .$

$By Baye' s theorem, P (E | F) = \frac{P (E) P (F | E)}{P (E) P (F | E) + P (E^{C}) P (F | E^{c})} = \frac{0.005 \times 0.95}{0.005 \times 0.95 + 0.995 \times 0.1} = \frac{0.00475}{0.10425} = 0.0455 Hence, the required probability is 0.0455 .$

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