A company comes up with a new brain scanning machine that can predict tumors wit

Question

A company comes up with a new brain scanning machine that can predict tumors with very high sensitivity of 0.99. The specificity of the device is 0.8. Suppose that, in the general population, the probability of having a brain tumor is 0.001.
(a) What is the probability that a random person from the population will actually test positive in this machine?
(b) What is the probability that a random person from the population will actually test negative in this machine?
(c) If a person tests positive using this machine, what is the probability that the person actually had a brain tumor (PV+)?
(d) If a person tests negative using this machine, what is the probability that the person actually did not have a brain tumor (PV-)

Explanation / Answer

In the above context, the definiton of sensitivity and specificity will be:

sensitivity = people actually having tumors are tested positive in the machine.

specificity = people actually not having tumors are tested negative in the machine.

(a) P = (Phaving a brain tumor * Sensitivity) + (Pnot having a brain tumor * (1-Specificity))

= (0.001 * 0.99) + (0.999*0.2) = 0.00099 + 0.1998 = 0.20079

(b) P = (Phaving a brain tumor * (1-Sensitivity) + (Pnot having a brain tumor * Specificity)

=(0.001*0.01) + (0.999*0.8) = 0.00001 + 0.7992 = 0.79921

(c) It is actually the definiton of sensitivity i.e. people actually having tumors are tested positive in the machine.

P = 0.99

(d) It is actually the definiton of specificity i.e. people actually not having tumors are tested negative in the machine.

P = 0.8

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