Download 211_Section7_6problem

Section 7.6
Bayes’ Theorem and Applications
We can also relate tree diagrams to having a disease and how a
person/item tested. If they have a disease, does the test also
support that result?
New terms:
Testing Positive: means that the test thinks that the person/item
has the disease.
Testing Negative: means that the test thinks that the person/item
does not have the disease.
Test
Condition
Has the
Disease
Doesn’t have
the Disease
Positive
Negative
Event
True Positive
False Negative
Positive
False Positive
Negative
True Negative
Four Disease Conditions
True Positive: The test states that the person/item has the disease
when they do really have the disease
False Negative: The test states that the person/item does not have
the disease when they do have the disease
False Positive: The test states that the person/item has the disease
when they really do not have the disease
True Negative: The test states that the person/item does not have
the disease when they do not have the disease
Example 1: Problem 3.2.7 from Statistics for Life Sciences
Suppose that a medical test has a 92% chance of detecting a
disease if the person has the disease and a 94% chance of
correctly indicating that the disease is absent if the person really
does not have the disease. Suppose that 10% of the population has
the disease.
1. What is the probability that a randomly chosen person will
test positive?
2. Suppose that a randomly chosen person does test
positive. What is the probability that this person really has
the disease?
Let D be the event that the person has the disease.
Let P be the event that the person tests positive.
We will need to use some equations that we have learned about
this previously
P( A  B)  P( A)  P ( B A) or
P( A  B)  P( B)  P( A B)
Pr A B   Pr(B)  Pr( A and B) and
PrB A  Pr( A)  Pr( A and B)
P( B)  P( B A)  P( A)  Pr(B A )  Pr( A )
This problem illustrates the use of Bayes’ Theorem and
Applications:
Bayes’ Theorem (Short Form)
If A and T are events, then
Bayes Formula
P(T A) P( A)
P( A T ) 
P(T A) P( A)  P(T A) P( A)
P( A  T )
P( A  T )  P( A  T )
P( A T ) 
Using a Tree
P( A T ) 
P( Using A and T branches)
Sum of P( Using branches ending in T )
Example 2
Use Bayes’ theorem or a tree diagram to calculate the indicated
probability.
3. P( X Y )  0.8 , P (Y )  0.3, P( X Y )  0.5 , Find P (Y X )
Bayes’ Theorem (Expanded Form)
If the events A1, A2 , and A3 form a partition of the sample space S ,
then
P(T A1 ) P( A1 )
P( A1 T ) 
or
P(T A1 ) P( A1 )  P(T A2 ) P( A2 )  P(T A3 ) P( A3 )
P( A1 T ) 
P( A1  T )
P( A1  T )  P( A2  T )  P( A3  T )
Example 3
Use Bayes’ theorem or a tree diagram to calculate the indicated
probability.
3. Y1, Y2 , Y3 form a partition of S . P( X Y1 )  0.2 , P( X Y2 )  0.3 ,
P( X Y3 )  0.6 , P(Y1 )  0.3 , P(Y2 )  0.4 , Find P (Y1 X )
18. Professor Frank Nabarro insists that all senior physics majors
take his notorious physics aptitude test. The test is so tough that
anyone not going on to a career in physics has no hope of passing,
whereas 60% of the seniors who do go on to a career in physics
still fail the test. Further, 75% of all senior physics majors in fact go
on to a career in physics. Assuming that you fail the test, what is
the probability that you will not go on to a career in physics?
24. In 2000, 59% of all Caucasians in the U.S., 57% of all AfricanAmericans, 58% of all Hispanics, and 54% of residents not
classified into one of these groups used the Internet to search for
information. At that time, the U.S. population was 69% Caucasian,
12% African-American, and 13% Hispanic. What percentage of
U.S. residents who used the Internet for information search were
African-American?
26. A New York Times survey showed that 51% of those who had
no preschool education were arrested or charged with a crime by
the time they were 19, whereas only 31% who had preschool
education wound up in this category. The survey did not specify
what percentage of the youths in the survey had preschool
education, so let us take a guess at that and estimate that 20% of
them had attended preschool. What percentage of the youths
arrested or charged with a crime had no preschool education?