Chapter 15 Probability Trees, Odds and Probability
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Transcript Chapter 15 Probability Trees, Odds and Probability
Chapters 14, 15 (part 2)
Probability Trees, Odds
i) Probability Trees: A
Graphical Method for
Complicated Probability
Problems.
ii) Odds and Probabilities
Probability Tree Example: probability
of playing professional baseball
6.1% of high school baseball players play college
baseball. Of these, 9.4% will play professionally.
Unlike football and basketball, high school players can
also go directly to professional baseball without playing
in college…
studies have shown that given that a high school player
does not compete in college, the probability he plays
professionally is .002.
Question 1: What is the probability that a high school
baseball player ultimately plays professional baseball?
Question 2: Given that a high school baseball player
played professionally, what is the probability he played in
college?
Question 1: What is the probability that a high school
baseball player ultimately plays professional baseball
Play prof. .094
.061*.094=.005734
Play coll 0.061
.906
HS BB Player
Play prof. .002
Does not play coll
0.939
Does not Play
prof. .998
.939*.002=.001878
P(hs bb player plays professionally)
= .061*.094 + .939*.002
= .005734 + .001878
= .007612
Question 2: Given that a high school baseball player played
professionally, what is the probability he played in college?
Play prof. .094
Play coll 0.061
.906
.061*.094=.005734
P(hs bb player plays professionally)
= .005734 + .001878
= .007612
HS BB Player
Play prof. .002
.939*.002=.001878
Does not play coll
0.939
Does not Play
prof. .998
P(played in college given that played professionally)
.005734
=
.7533
.007612
Example: AIDS Testing
V={person has HIV}; CDC: Pr(V)=.006
P : test outcome is positive (test
indicates HIV present)
N : test outcome is negative
clinical reliabilities for a new HIV test:
1. If a person has the virus, the test result will
be positive with probability .999
2. If a person does not have the virus, the test
result will be negative with probability .990
Question 1
What is the probability that a randomly
selected person will test positive?
Probability Tree Approach
A probability tree is a useful way to
visualize this problem and to find the
desired probability.
Probability Tree
clinical
reliability
clinical
reliability
Multiply
branch probs
Question 1: What is the probability that a
randomly selected person will test positive?
Pr( P ) .00599 .00994 .01593
Question 2
If your test comes back positive, what is
the probability that you have HIV?
(Remember: we know that if a person
has the virus, the test result will be
positive with probability .999; if a person
does not have the virus, the test result
will be negative with probability .990).
Looks very reliable
Question 2: If your test comes back positive, what is
the probability that you have HIV?
Pr( P ) .00599 .00994 .01593
P (have HIV given that test is positive)
.00599
=
.376
.00599 .00994
Summary
Question 1:
Pr(P ) = .00599 + .00994 = .01593
Question 2: two sequences of branches
lead to positive test; only 1 sequence
represented people who have HIV.
Pr(person has HIV given that test is positive)
=.00599/(.00599+.00994) = .376
Recap
We have a test with very high clinical
reliabilities:
1. If a person has the virus, the test result will be
positive with probability .999
2. If a person does not have the virus, the test result
will be negative with probability .990
But we have extremely poor performance when
the test is positive:
Pr(person has HIV given that test is positive) =.376
In other words, 62.4% of the positives are false
positives! Why?
When the characteristic the test is looking for is
rare, most positives will be false.
ODDS AND
PROBABILITIES
World Series Odds
From probability to odds
From odds to probability
From Probability to Odds
If event A has
probability P(A), then
the odds in favor of A
are P(A) to 1-P(A). It
follows that the odds
against A are 1-P(A)
to P(A)
If the probability the
Boston Red Sox win
the World Series is
.20, then the odds in
favor of Boston
winning the World
Series are .20 to .80
or 1 to 4. The odds
against Boston
winning are .80 to .20
or 4 to 1
From Odds to Probability
If the odds in favor
of an event E are a
to b, then
P(E)=a/(a+b)
If the odds against
an event E are c to
d, then
P(E’)=c/(c+d)
(E’ denotes the
complement of E)
E = win World Series
Team
Odds against
winning
P(E’)=Prob of
not winning
RED SOX
4/1
4/5=.80
DODGERS
5/1
5/6=.833
TIGERS
5/1
5/6=.833
CARDINALS
11/2
11/13=.846
BRAVES
7/1
7/8=.875
A’s
15/2
15/17=.882
TB RAYS
14/1
14/15=.933
INDIANS
14/1
14/15=.933
REDS
16/1
16/17=.941
PIRATES
16/1
16/17=.941