Cancer Diagnosis: Interpreting X-ray Results with Bayes' Theorem

1. To find the probability that the patient has cancer given a positive x-ray result, we can use Bayes' theorem:

P(Cancer|Positive) = (P(Positive|Cancer) * P(Cancer)) / P(Positive)

where P(Positive|Cancer) is the probability of a positive x-ray given the patient has cancer, P(Cancer) is the probability of the patient having cancer, and P(Positive) is the overall probability of a positive x-ray result.

P(Positive|Cancer) = 0.696 (given in the problem) P(Cancer) = 0.01 (1% or 0.01 probability of having cancer based on the given information) P(Positive) = (P(Positive|Cancer) * P(Cancer)) + (P(Positive|Non-cancer) * P(Non-cancer))

P(Positive|Non-cancer) = 0.193 (given in the problem) P(Non-cancer) = 0.99 (99% or 0.99 probability of not having cancer based on the given information)

P(Positive) = (0.696 * 0.01) + (0.193 * 0.99) = 0.00696 + 0.19107 = 0.19803

Now, we can calculate the probability of having cancer given a positive x-ray result:

P(Cancer|Positive) = (0.696 * 0.01) / 0.19803 = 0.00696 / 0.19803 = 0.0351

Therefore, the probability that the patient has cancer given a positive x-ray result is approximately 0.0351 or 3.51%.

2. If the doctor assumes that cancerous and non-cancerous lesions are equally likely (ignoring the base rate of cancer), the mistaken conclusion drawn from the test would be that the patient has a high probability of having cancer. This is because the doctor would only consider the positive x-ray result (0.696) without considering the low base rate of cancer (0.01). This can lead to unnecessary worry, anxiety, and potentially unnecessary treatments.

3. It is important to have hospital procedures that require additional tests to be performed before a patient undergoes treatment in order to minimize the chance of misdiagnosis or unnecessary treatments. In this case, a positive x-ray result alone is not sufficient to confirm the presence of cancer, as there is still a relatively low probability (3.51%) that the patient actually has cancer. Additional tests, such as a biopsy or further imaging, can provide more accurate information and help confirm the diagnosis before initiating any treatments. This helps to ensure that the patient receives the appropriate and necessary care based on a more comprehensive evaluation.