Chapter 1: Sets and Probability
Section 1.6-1.7
Conditional Probability
Let E and F be two events in a sample space S. The
probability that E occurs given that F has occurred is defined to be
S
E
F
F
E∩F
The Old Sample Space
The New Sample Space
Example 1 Let P (E) = 0.7, P (F ) = 0.3, and P (E ∩ F ) = 0.1. Find
a) P (E|F )
b) P (F C |E)
c) P (F |E C )
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d) P (E ∪ F |F )
e) P (E C ∩ F |E)
Example 2 A pair of fair six-sided dice is rolled.
a) What is the probability that a 2 is rolled (on either one of the dice)?
b) What is the probability that the sum is less than 5 given that a 2 is rolled?
c) What is the probability that a 2 is rolled given that the sum is less than 5?
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Example 3 The table below gives the number of students of each classification who are
majoring and not majoring in business in a class of 110 students.
Freshmen
Business
10
Non-Business
8
Total
18
Sophomores Juniors Seniors Total
17
20
12
59
3
15
25
51
20
35
37
110
A student is randomly selected from this class. Find the probability of the following events.
a) A student, who is a business major, is a freshman.
b) A freshman is a business major.
c) An underclassman given that the student is not a business major.
Product Rule: If E and F are two events in a sample space S with P (E) > 0 and
P (F ) > 0, then
Note: This is just a rearrangement of the formula for conditional probability.
If we have a sequence of experiments in which the outcomes and associated probabilities of
each experiment depend on the outcomes of the preceeding experiments, we can represent
.
this by a
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Example 4 A jar contains 3 red marbles and 5 green marbles. Two marbles are selected
from the jar, one at a time and without replacement, and the color of each marble is
observed.
a) What is the probability that the first marble picked is green?
b) What is the probability that the second marble picked is red given that the first marble
is green?
c) What is the probability that the second marble picked is red?
d) What is the probability that the first marble picked is green given that the second marble
is red?
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Example 5 (Tomastic/Epstein) Two machines turn out all the products in a factory, with
the first machine producing 40% of the product and the second 60%. The first machine
produces defective products 2% of the time and the second machine 4% of the time.
a) What is the probability that a defective product is produced at this factory given that
it was made on the first machine?
b) What is the probability that a defective product is produced at this factory?
c) Given a defective product, what is the probability it was produced on the first machine?
Example 6 In a survey of 1000 eligible voters selected at random, it was found that
80% had a college degree. Additionally, it was found that 80% of those who had a college
degree voted in the last presidential election, whereas 55% of the people who did not have
a college degree voted in the last presidential election.
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Assuming the poll is representative of all eligible voters, find the probability that an eligible
voter selected at random
a) had a college degree and voted in the last presidential election.
b) did not have a college degree and did not vote in the last presidential election.
c) voted in the last presidential election.
if
Two events E and F are said to be
Note: E and F are independent if the outcome of one does not effect the outcome of
the other
Independent Events Theorem: Let E and F be two events with P (E) > 0 and
P (F ) > 0. Then E and F are independent if and only if
Note: Do NOT confuse independence with two events being mutually exclusive!
Example 7 Determine if the given events E and F are independent.
a) P (E) = 0.5, P (F ) = 0.6, and P (E ∩ F ) = 0.4
b) P (E ∩ F C ) = 0.3, P (E ∩ F ) = 0.2, and P (F ∩ E C ) = 0.2
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Example 8 Assume the events E and F are independent with P (E) = 0.3 and P (F ) =
0.6. Find P (E ∪ F ).
Example 9 A fair coin is flipped three times. Let E be the event “at most one head” and
F be the event “at least one head and at least one tail.” Are these two events independent?
A set of events {E1 , E2 , ..., En } is said to be
if, for any k of these events, the probability of the intersection of these k events is the
product of the probabilities of each of the k events. This must hold for any k = 2, 3, ..., n.
Example 10 For the set of events {E, F, G} to be independent, all of the following must
be true:
• P (E ∩ F ) = P (E)P (F )
• P (E ∩ G) = P (E)P (G)
• P (F ∩ G) = P (F )P (G)
• P (E ∩ F ∩ G) = P (E)P (F )P (G)
Note: If two events E and F are independent, then so are E and F C , E C and F , and
E C and F C . Similar statements are true about a set of events.
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