A research laboratory develops a new serologic test for detecting prostate cancer. The new assay is compared to transrectal ultrasound-guided prostate biopsy to determine sensitivity and specificity parameters. It is found that the test result is negative in 95% of patients who do not have the disease. If the new assay is used on 8 blood samples taken from patients without prostate cancer, what is the probability of all 8 test results coming back negative?

Question

A research laboratory develops a new serologic test for detecting prostate cancer.  The new assay is compared to transrectal ultrasound-guided prostate biopsy to determine sensitivity and specificity parameters.  It is found that the test result is negative in 95% of patients who do not have the disease.  If the new assay is used on 8 blood samples taken from patients without prostate cancer, what is the probability of all 8 test results coming back negative?

Quizplus · Accepted Answer

Each of the 8 blood sample results is an independent event (ie, one patient's result has no impact on another's) with a 0.95 (95%) probability of correctly testing negative and a 0.05 (5%) probability of incorrectly testing positive. To calculate the chance of all 8 tests testing negative, multiply the probability of each test returning a negative result: (Choices A and B) The value 0.05 × 8 is equivalent to adding the probability of a sample testing positive 8 times, and 0.95 × 8 is equivalent to adding the probability of a sample testing negative 8 times. However, the combined probability for a series of independent events is calculated using multiplication, not addition. For instance, when flipping a coin twice the probability of getting 2 heads in a row is not 0.5 + 0.5 = 0.5 × 2 = 1 (100%) but rather 0.5 × 0.5 = 0.25 (25%). (Choice C) The probability of all 8 samples incorrectly testing positive is given by 0.05⁸. (Choices E and F) Overall, 2 outcomes are possible: either all 8 samples will return the same test result or at least 1 sample will test differently. Because the total probability must add up to 1 (100%) and the probability that all 8 samples will return negative is given by 0.95⁸, the probability of at least 1 sample testing positive is 1 - 0.95⁸. Conversely, the probability that all 8 samples will return positive is 0.05⁸ while the probability of at least 1 sample testing negative is 1 - 0.05⁸. Educational objective: If events are independent, the probability that all events will turn out the same is the product of the separate probabilities for each event. The probability of at least 1 event turning out differently is given as 1 - P(all events being the same).

A Research Laboratory Develops a New Serologic Test for Detecting