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I am studying Bayes's rule in the Mathematical Statistics and Data Analysis by John A. Rice (3rd edition) and had a question about the example from the book.

Conditions:

  • In the absence of any special information, the probability that a woman (of the age and health status of this patient) has breast cancer is 1%.
  • If the patient has breast cancer, the probability that the radiologist will correctly diagnose it is 80%.
  • If the patient has a benign lesion (no breast cancer), the probability that the radiologist will incorrectly diagnose it as cancer is 10%.

The question is, "What is the probability that a patient with a positive mammogram actually has breast cancer?"

Using Bayes' rule, I have below. enter image description here How would you get the probability of 'correct diagnose given no breast cancer'? The correct answer to this example is 0.075 (7.5%), and I am not sure how they got to this number.

learner
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  • "If the patient has a benign lesion (no breast cancer), the probability that the radiologist will incorrectly diagnose it as cancer is 10%." – Henry Sep 26 '21 at 01:03
  • Incidentally, your handwritten notes are unclear as to whether $\oplus$ means "correctly diagnose" or "diagnose cancer" but $\frac{0.008}{0.008+0.099}\approx 0.075$ – Henry Sep 26 '21 at 01:12
  • Related, possible duplicate. See the link there too. https://math.stackexchange.com/questions/2279851/applied-probability-bayes-theorem/2279888#2279888 – Ethan Bolker Sep 26 '21 at 01:14
  • @Henry Why did you use 0.1 for the P(correctly diagnose | No breast cancer)? I think what you did is correct, but seeing the P(incorrectly diagnose | No breast cancer) = 0.1, I'm not sure how why P(correctly diagnose | No breast cancer) should also be 0.1 – learner Sep 26 '21 at 04:15
  • I did not "use 0.1 for the P(correctly diagnose | No breast cancer)". I used $0.1$ for $P(\text{diagnose breast cancer} \mid \text{No breast cancer})$ and multiplied it by $P(\text{No breast cancer}) =0.99$ to give $0.099$ – Henry Sep 26 '21 at 17:39
  • Your error is mixing up "correct diagnosis" and "positive result" by using the same notation "$+$" for both. Here's an answer to a similar question as yours. – ryang Dec 09 '21 at 05:09

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The statement says: "If the patient has a benign lesion (no breast cancer), the probability that the radiologist will incorrectly diagnose it as cancer is 10%". This means the person does not have cancer, but the radiologist diagnoses the opposite. Using your notation: P (+ | No Cancer) = 0.1

$P(C | +) = \frac{P(+ | C)*P(C)}{P(+ | C)*P(C) + P(+ | NC)*P(C)}$

$P(C | +) = \frac{0.08*0.01}{0.08*0.01 + 0.1*0.99}$

$P(C | +) = 0.0747$

Alejandro González Tapia
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