A patient is tested for cancer. This type of cancer occurs in 5% of the population....

A patient is tested for cancer. This type of cancer occurs in 5% of the population. The patient has undergone testing that is 90% accurate and the results came back positive. What is the probability that the patient actually has cancer? (Conditional probability)

Expert Solution

P[ Person has cancer ] = 5% = 0.05

P[ Person does not have cancer ] = 1 - 0.05 = 0.95

P[ Person has cancer deducted | he has cancer ] = 90% = 0.9

P[ Person has cancer deducted | he does not have cancer ] = 1 - 0.9 = 0.1

P[ the test is positive ] = P[ Person has cancer ]*P[ Person has cancer tested positive | he has cancer ] + P[ Person does not have cancer ]*P[ Person has cancer tested positive | he does not have cancer ]

P[ the test is positive ] = 0.05*0.9 + 0.95*0.1

P[ the test is positive ] = 0.045 + 0.095

P[ the test is positive ] = 0.14

P[ patient has cancer | the test is positive ] = P[ patient has cancer and the test is positive ]/P[ the test is positive ]

P[ patient has cancer | the test is positive ] = P[ Person has cancer ]*P[ Person has cancer tested positive | he has cancer ] /P[ the test is positive ]

P[ patient has cancer | the test is positive ] = 0.05*0.9 / 0.14

P[ patient has cancer | the test is positive ] = 0.045/0.14

P[ patient has cancer | the test is positive ] = 0.3214

orchestra answered 1 year ago

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