Example 18 - Suppose reliability of a HIV test is: people - Bayes theoram

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Example 18 Suppose that the reliability of a HIV test is specified as follows: Of people having HIV, 90% of the test detect the disease but 10% go undetected. Of people free of HIV, 99% of the test are judged HIV –ive but 1% are diagnosed as showing HIV +ive. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV + ive. What is the probability that the person actually has HIV? Let E : person selected has HIV F : person selected does not have HIV G: test judges HIV +ve We need to find the Probability that the person selected actually has HIV, if the test judges HIV +ve i.e. P(E|G) P(E|G) = 𝑃 𝐸﷯ . 𝑃(𝐺|𝐸)﷮𝑃 𝐸﷯ .𝑃 𝐺|𝐸﷯+𝑃 𝐹﷯ . 𝑃(𝐺|𝐹)﷯ Putting values in formula, P(E|G) = 0.001 × 0.9﷮0.001 × 0.9 + 0.999 × 0.01﷯ = 9 × 10﷮ − 4﷯﷮9 × 10﷮ − 4﷯ + 99.9 × 10﷮ − 4﷯﷯ = 10﷮ − 4﷯ × 9﷮ 10﷮ − 4﷯ [9 + 99.9]﷯ = 9﷮108.9﷯ = 𝟗𝟎﷮𝟏𝟎𝟖𝟗﷯ = 0.083 (approx) Therefore, required probability is 0.083

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