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And plug in the following data:
P(cancer) = 2.6% (260 out of 10,000 have cancer)
P(~cancer) = 97.4% (9740 out of 10,000 have no cancer)
P(positive x-ray | cancer) = 85%
(85% of people with lung cancer have positive x-ray)
P(positive x-ray | ~cancer) = 6%
(6% of people without lung cancer have positive x-ray)
And plug in the above data into the above Bayes' theorem, we will get:
P(cancer | positive x-ray) = 85%* 2.6% / (85%*2.6%+6%*97.4%)
= 0.0221 / (0.0221+0.0584)
= 0.0221 / 0.0805
= 0.274
As you can see, comparing to the non-risky population (the probability of having cancer
0.028), the probability value of 0.274 of a person in the risky group is much higher. This makes
sense since the prior probability of getting lung cancer is higher in this high risk group. In this
new example, the quality of the x-ray equipment does not change. The only thing changed is
the prior cancer rate, from 0.2% to 2.6%. At first look to the new problem, most people will
give the same wrong answer of 85%. But Bayes' reasoning gives us more objective and correct
answer. Here is an example that computer reasoning can be better than a human!
Bayes' reasoning can be used in situations that have multiple evidences. Let's use Example
2, which is the extension of Example 1, to illustrate how this is done.
Example 2: “Lung cancer is the leading cause of cancer death in the United States.”
(Williams, 2003, p. 463) Suppose that about 0.2% of the population living in US with age
above 20 has lung cancer. When doing an annual check, assume that 85% of the people with
lung cancer will show positive for the chest x-ray test. On the other hand, chest x-ray will
have false alarms: 6% of the people without lung cancer will also show positive for the chest
x-ray test. Suppose that a hospital will do two lung cancer screen tests for each annual check
patient (assume the two tests are independent). The second test called CT scan is done to
improve the accuracy of diagnosis. Suppose that the CT scan has the following
characteristics: it returns positive for 85% of the people with lung cancer; it has a lower false
rate than the x-ray test and will return false positive for one out of one thousand people
without lung cancer. If a person went through the annual check and had positives on both
the chest x-ray and the CT scan, what is the probability that he/she has the lung cancer?
Answer: We can solve this problem by using the Bayes' theorem twice. We already know
that the probability of a person has cancer given that he has positive x-ray is 2.8%; the
probability of a person has no cancer given that he has positive x-ray is 97.2%. We can use
this result and continue to solve the problem as follows:
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