Question

Medical tests are frequently used to decide whether a person has a particular disease. The sensitivity of a test is the probability that a person having the disease will test posi- tive; the specificity of a test is the probability that a person not having the disease will test negative. In the Providence Alaska Medical Center, a test for a certain disease has been used for years. Experience with the test indicates that its sensitivity is 0.934 and that its specificity is 0.968. Furthermore, it is known that roughly 1 in 500 adults has this disease. (a) Construct a tree diagram containing the given probabilities. (b) Suppose an adult gets a positive test result. What is the probability that this adult in- deed has the disease? (c) Suppose an adult gets a negative test result. What is the probability that this adult ac- tually has the disease?

Answer 1

(b) the probability that this adult in- deed has the disease will be 0.0286

(c) The probability that this adult ac- actually has the disease after negative test result will be: 0.0007

(a) The tree diagram containing the probabilities is attached below.

(b) The probability that this adult in- deed has the disease will be:

$P(\text { d } \mid \text { positive })=\frac{0.002 \times 0.968}{0.002 \times 0.968+0.998 \times 0.066}=0.0286$

(c) The probability that this adult ac- actually has the disease after negative test result will be

$\begin{aligned}P(d | \text { negative })= & \frac{0.002 \times 0.032}{0.002 \times 0.032+0.998 \times 0.934} \\= & 0.00007\end{aligned}$

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