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Question

A laboratory blood test is 99% effective in detecting a certain diesease when it is fact present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e., if a healthy person is tested, then with probability 0.005, the test will imply he has the disease). If 0.1% of the population actually has the disease, what is the probability that a person has diease given that his test result is positive?


Solution

Let E1 : the event that the person has disease and E2 : the events that the person is healthy.

Then, E1 and E2 are mutually exclusive and exhaustive. .

Moreover, P E1=0.1% = 0.1100=0.001 and P E2=1-0.001=0.999

Let E: the event that test is positive, 

P(EE1)= P(result is positive given the person has disease)

=99% 99100 =0.99

Probability that a person does not have disease and test result is positive. 

P(EE2)=0.5 

By using Baye's theorem, we obtain

P(E1E)=P(EE1)P(E1)P(EE1)P(E1)+P(EE2)P(E2)

=0.001×0.990.001×0.99+0.999×0.005=0.0000990.00099+0.004995=0.000990.005985=9905985=110665=22133

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