Question

In: Statistics and Probability

A scientist designed a medical test for a certain disease. Among 100 patients who have the...

A scientist designed a medical test for a certain disease. Among 100 patients who have the disease, the test will show the presence of the disease in 96 cases out of 100, and will fail to show the presence of the disease in the remaining 4 cases out of 100. Among those who do not have the disease, the test will erroneously show the presence of the disease in 3 cases out of 100, and will show there is no disease in the remaining 97 cases out of 100.

What is the probability that a patient who tested positive on this test actually has the disease, if it is estimated that 20% of the population has the disease?

	A.	0.9074 or 90.74%
	B.	0.04 or 4%
	C.	0.96 or 96%
	D.	0.9407 or 94.07%

Expert Solution

P[ A person has disease ] = 20% = 0.2

P[ A person does not have disease ] = 1 - P[ A person has disease ] = 1 - 0.2 = 0.8

P[ the test is positive | Patient actually has the disease ] = 96/100 = 0.96

P[ the test is positive | Patient does not have the disease ] = 3/100 = 0.03

P[ Patient actually has the disease | the test is positive ] = P[ Patient actually has the disease and the test is positive ] / P[ the test is positive ]

P[ the test is positive ] = P[ Patient actually has the disease and the test is positive ] + P[ Patient does not the disease and the test is positive ]

P[ Patient actually has the disease and the test is positive ] = P[ the test is positive | Patient actually has the disease ]*P[ A person has disease ]

P[ Patient actually has the disease and the test is positive ] = 0.96*0.2

P[ Patient actually has the disease and the test is positive ] = 0.192

P[ Patient does not the disease and the test is positive ] = P[ the test is positive | Patient does not have the disease ] *P[ A person does not have disease ]

P[ Patient does not the disease and the test is positive ] = 0.03*0.8

P[ Patient does not the disease and the test is positive ] = 0.024

P[ the test is positive ] = 0.024 + 0.192

P[ the test is positive ] = 0.216

P[ Patient actually has the disease | the test is positive ] = 0.192/0.216

P[ Patient actually has the disease | the test is positive ] = 0.8889

orchestra answered 2 years ago

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