Question

2) Suppose that, in the course of a regular check-up, a doctor discovers that the patient has a potentially cancerous lesion. Most lesions are benign (non-cancerous), say 99%. The doctor orders an x-ray just in case. In laboratory tests on malignant (cancerous) lesions, the x-ray returns positive (cancer-affirming) results 79.2% of the time and negative results 20.8% of the time. In laboratory tests on benign lesions, the x-ray returns positive results only 9.6% of the time and negative results 90.4% of the time.

(a) The patient’s x-ray comes back positive. What is the probability that the patient has cancer?

(b) Suppose that the doctor calculates the probability that the patient has cancer without regard to the base rate of cancer in the population – that is, the doctor uses Bayes’ Rule but assumes that cancerous and non-cancerous lesions are equally likely. What mistaken conclusion will the doctor draw from the test? Explain why it is important in these situations to have hospital procedures that require additional tests to be performed before a patient undergoes treatment.

Answer 1

Answer (a) : To find out the probability, we need to figure out the number of events and the number of outcomes in the complete scenario where the patient is being diagnosed and a report is being sought. Here, the number of events is 3 (Three). First one being the case of benign result which is 99%, the second one being the case of malignant lesions which is 79.2%, the third case is where the X-ray being positive or negative, which is 9.6% positive. Here, the number of outcomes is 6.

Therefore, the probability of the patient having cancer= Number of events / number of outcomes

                                                                                                  = 3/6

                                                                                                  =0.50 or 50%

  Answer (b) : If the Doctor uses Bayes’ Rule and assumes that cancerous and non-cancerous lesions are equally likely, it will lead to an erroneous conclusion. This will depend on the laboratory tests and the further investigations and tests performed on the patient. A lot of factors instigate the development of such a malignant disease. Therefore, simply assumption of the likelihood of the positiveness of the result that the patient is affected by the disease, is not the appropriate method.

                                            If on the basis of assumption, the complete treatment procedure of the malignant disease is started, it will imply that the patient will have to undergo several treatment procedures which will expose the patients physic to those medicines or drugs which are specifically meant for the malignant disease affected patients. If this patient is not affected by the malignant disease, then the intake of such drugs and medicines will negatively impact the health of the patient, thus making him prone to this or other diseases or disorders. Therefore, additional tests and procedures must be undergone before starting with the treatment process of the patient.