Information Technology Reference
In-Depth Information
1.
We use the Bayes' theorem:
│−&
=
P(|)∗()
P
(
|
)
∗
(
)
+
(
|~
)
∗(~)
2.
And plug in the following data:
P(cancer new prior) = 2.8% (the poster probability of )
(9)
P(~cancer new prior) = 97.2% (the complement of equation (9))
(10)
P(positive CT scan | cancer) = 85%
(85% of people with lung cancer have positive CT scan)
(11)
P(positive CT scan | ~cancer) = 0.1%
(0.1% of people without lung cancer have positive CT scan)
(12)
3.
And plug in the above data into the above Bayes' theorem, we will get:
P(cancer | positive x-ray & positive CT scan) =
85%* 2.8% / (85%*2.8%+0.1%*97.2%)
= 0.0238 / (0.0238+0.00097)
= 0.0238 / 0.02477
= 0.96
As you can see, the person's probability of having lung cancer is very high in this instance.
In this example, each application of the Bayes' theorem can be viewed as a mapping from
one statistical sample space to another statistical sample space and there are two such
mappings as shown in Figure 3.
In Figure 3, P(xp&c) means the probability of a person who has lung cancer and is x-ray
positive; P(xp&CTp&c) means the probability of a person who has lung cancer and is both
x-ray positive and CT scan positive. Similarly, P(xp&CTp&h) means the probability of a
person who is healthy and is both x-ray positive and CT scan positive. To help our
understanding of what's going on, we list some calculated data below (assume total of
10,000 people):
