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Math 141H-copyright Joe Kahlig, 15C
Section 7.5: Conditional Probability and Independence and Section 7.6: Bayes’ Theorem
Definition:
Conditional probability is the probability of an event, A, will occur given that
another event, B, has already happened. This is denoted as P (A|B).
Example: This table classifies the English, History, Math, and Poly Sci majors at State U according
to their year. (There are no double majors.)
Freshmen(F)
Sophomores(Soph)
Juniors(J)
Seniors(Sr)
Totals
English(E)
64
35
31
41
171
History(H)
55
41
33
52
181
Math(M)
29
32
50
69
180
Poly Sci(PS)
70
33
41
37
181
Totals
218
141
155
199
713
If a student is selected at random, find
A) P(H | soph) =
B) P( (soph ∪ J) |PS) =
Interpret these Statements: Express these questions using the correct probability notation.
A factory makes widgets on three different machines; A, B and C; and a widget is selected at random.
1. What is the probability that it is defective knowing that the widget was made on machine C?
2. If the widget was made on machine A, what is the probability that it is defective?
3. What is the probability it is made on machine A and is not defective?
4. What is the probability the widget is selected from line B or is defective?
5. What is the probability that the defective widget was made on machine B?
6. What is the probability that the widget is defective and came from machine A or machine B?
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Math 141H-copyright Joe Kahlig, 15C
Example: Use the probability distribution and the events to answer these questions.
S
a
b
c
d
e
f
prob
0.15
0.08
0.21
0.12
0.25
0.19
E = { a, c, d, e}
F = {b, d, f}
G = {a, b, d}
Compute
A) P (E|G) =
B) P (G|F ) =
Definition: Let E and F be two events of the sample space S. Then the conditional probability that
E occurs given that F has occurred is defined as
P (E|F ) =
Example: A Jar has 5 red, 4 green, and 3 yellow items. Two items are drawn in succession without
replacement. Construct the probability tree that represents this experiment.
Math 141H-copyright Joe Kahlig, 15C
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Example: Draw the tree that represents this experiment: Draw one ball from box A and place it into
box B. Then draw one ball from box B.
Box A
3 red
5 green
Box B
2 red
5 yellow
Example: You have two jars. Jar 1 had 4 red, 3 green and 2 black balls. Jar 2 has 8 red and 5 green
balls. The experiment is to draw a single ball. We have been told that a person is twice as likely to
draw an item from Jar 1 as from Jar 2.
A) Draw the probability tree for this problem.
B) P (r|J1) =
P (b|J2) =
C) P (J2 ∩ r) =
D) P (g) =
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Math 141H-copyright Joe Kahlig, 15C
Example: Use fact that P (C ∩ D) = 0.09 and the tree to answer these questions.
0.7
A) P (B|D) =
D
A
E
0.4
0.8
D
B
E
D
0.1
B) P (A ∪ D) =
C
E
Example: Based on data from a dental survey, it has been determined that 42% of 12-year olds have
never had a cavity, 34% of 13-year olds have never had a cavity, and 28% of 14-year olds have never
had a cavity. A child is selected at random from a group of students of which 28% of them are 12-year
olds, 45% of them are 13-year olds and the rest are 14-year olds.
A) What is the probability that
the selected student has a cavity?
B) What is the probability that the student,
who had a cavity, was 14 years old?
Math 141H-copyright Joe Kahlig, 15C
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Example: A medical test has been designed to detect the presence of a certain condition. Among
those who have the condition, the probability that the test will detect it is 98%. The test will give a
false positive 6% of the time. It is generally known that 12% of the population has the condition.
A) If the test administered to a person is positive,
what it the probability that the person actually
has the condition?
B) If the test is administered a second time and
it comes back positive both times, what is the
probability that the person has the condition?
Example: Draw cards from a deck of cards without replacement.
A) What is the probability that the
second card drawn is a heart.
B) P(3rd card is a heart) =
C) P(28th card is a heart) =
D) P(1st heart | 2nd is a club) =
E) P(1st heart | 2nd is a Queen) =
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Math 141H-copyright Joe Kahlig, 15C
Example: A computer store shipped 5 defective computers in its shipment of 30 computes to a local
school. The computers are arranged on desks inside the classroom. Find the probability that the
computer on desk 5 is defective if the computer on desk 1 and desk 10 are both good and the one on
desk 25 is defective.
Definition: Events E and F are said to be independent if
P (E|F ) = P (E)
and
P (F |E) = P (F )
Example: The following information was collected on 30 students.
Four of the students only wear glasses.
Twelve of the students only wear braces.
Six of the students wear glasses and braces.
The rest didn’t wear glasses and didn’t wear braces.
Determine whether the events G=“wear glasses” and B=“wear braces” are independent or not. Justify
your answer.
Math 141H-copyright Joe Kahlig, 15C
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Example: A kidney transplant doctor has three patients, A, B, and C, that will be receiving a kidney
transplant. The doctor has determined that patient A has a 12% chance of rejecting the kidney,
patient B has a 8% chance of rejecting the kidney, and patient C has a 21% chance of rejecting the
kidney.
A) What is the probability that none
of the patients reject the kidney.
B) What is the probability that exactly
one of the patients will reject the kidney.
Example: If P (A) = 0.4 and P (B) = 0.6, what is P (A ∩ B)?