You should have Studied Sections 4.5-4.8 from the texbook. Please report on what you studied from this assigned reading.

Report: I read 4.5-4.8



Complete the following problems from Chapter 4.


4.9 Suppose there is a medical diagnostic test for a disease. The sensitivity of the test is .95. This means that if a person has the disease, the probability that the test gives a positive response is .95. The specificity of the test is .90. This means that if a person does not have the disease, the probability that the test gives a negative response is .90, or that the false positive rate of the test is .10. In the popluation, 1% of the people have the disease. Let \(D\) be the event “the person has the disease” and let \(T\) be the event “the test gives a positive result.” What is the probability that a person tested has the disease, given the results of the test is positive? In other words, what is \(P(D|T)\)?

\(P(D|T) = \frac{P(T|D) * P(D)}{P(D) * P(T|D) + P(\tilde{D}) * P(T|\tilde{D})} = 0.8756\)


4.10 Suppose there is a medical screening procedure for a specific cancer that has sensitivity = .90, and specificity = .95. Suppose the underlying rate of the cancer in the population is .001. Let \(B\) be the event “the person has that specific cancer,” and let \(A\) be the event “the screening procedure gives a positive result.”

  1. What is the probability that a person has the disease given the results of the screening is positive?

\(P(D|T) = \frac{P(T|D) * P(D)}{P(T|D) * P(D) + P(T| \tilde{D}) * P( \tilde{D})} = 0.0177\)

  1. Does this show that screening is effective in detecting this cancer?

I answer yes and no (These are disjoint answers that span the universe, so I am guaranteed to have answered the question correctly!)

I say no because a person that tests positive has less than a 2% chance of actually having cancer. That sounds like a terrible rate.

That being said, the test is still quite effective. Notice that it will detect 90% of people that have cancer. We can see that \(P(D|T)\) is so low because only 0.1% of the population actually has cancer. The test appears bad because the small proportion of false positives (coming from the majority–99.9%–of the population) is bigger than the proportion of people that actually have cancer (0.1%).


4.11 and 4.12. You can complete either or both of these problems in place of either or both of 4.9 and 4.10. If you choose to complete these, please type them out as well as the solutions.