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x-ray test. If a person went through the annual check and had a positive chest x-ray, what is
the probability that he/she has the lung cancer? (For concreteness, you may assume that
there are 10,000 people participated the annual check)
Answer:
Most people will give the wrong answer of “the person will have 85% probability
of having the lung cancer.”
To get the answer right, we must first understand several important facts in statistics. The
first thing is that
P(A|X) ≠ P(X|A)
The reason that most people will get the incorrect answer of 85% is the confusion caused by
the above inequality relationship.
The correct answer for Example 1 is 2.8%. The following is the analysis and steps showing
how we get the correct answer:
1.
We start out by the basic probability definition:
P(cancer | positive x-ray) = number of people who have both cancer
and positive x-ray in the annual check / total number of people with
positive x-ray in the annual check
(Formula 2)
According to the meaning of conditional probability, the left side of Formula 2 is the
answer we are looking for.
Note:
the key of the above equation is to use the number of people who have both
cancer and positive x-ray as the numerator. If using people who have cancer as the
numerator, the result will be wrong since there are people who have cancer but have
negative x-ray test results.
2.
We use concrete number. Without losing generality, we assume there are 10,000 people
of age 20 and over participated in the annual check. Thus, we have the following data:
The number of people who have lung cancer in the annual check group is 10,000*0.2% = 20.
The number of people who are healthy in the annual check group is 10,000*99.8% = 9980.
The number of people who have lung cancer and have positive x-ray is 20*85% = 17.
The number of people who have lung cancer and have negative x-ray is 20*15% = 3.
The number of people who have no lung cancer and have positive x-ray is 9980*6% = 599.
3.
We use the data in step 2 and plug into the Formula 2 in step 1. We will get following
answer:
P(cancer | positive x-ray) = 17 / (17+599) = 17 / 616 = 0.028
Most people regard Bayes' theorem as statistical formula and overlook its reasoning logic.
We want to point out that it is also a reasoning method that captures the essence of
reasoning logic that reasons at the lower-levels. Thus, it is the theoretical foundation that
underpins the computer reasoning.
