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Importance of the Sample Space
„
Probability
„
Conditional Probability
and
Independence
The probability of an event depends on the
sample space in question. The sample space is
critical to determining the probability.
In certain types of probabilistic situations, the
entire sample space is not utilized – only a
portion is needed.
Rolling a Single Die
„
Consider rolling a single die.
„ List the sample space.
„ What is the probability of rolling a 5?
„ What is the probability of rolling a 5 given
that an odd number has been rolled?
Conditional Probability
„Let
E and F be events is a sample space S. The
conditional probability Pr(E | F) is the probability
of event E occurring given the condition that event F
has occurred. In calculating this probability, the
sample space is restricted to F.
Pr( E | F ) =
Pr( E ∩ F )
Pr( F )
provided that Pr(F) ≠ 0.
Example (continued)
Example: Conditional Probability
„
Twenty percent of the employees of Acme
Steel Company are college graduates. Of all
its employees, 25% earn more than $50,000
per year, and 15% are college graduates
earning more than $50,000. What is the
probability that an employee selected at
random earns more than $50,000 per year,
given that he or she is a college graduate?
„Let
H = "earns more than $50,000 per year" and
C = "college graduate."
„From the problem,
Pr(H) = .25
Pr(C) = .20
Pr(H ∩ C) = .15.
Therefore,
Pr( H | C ) =
Pr( H ∩ C ) .15 3
=
= .
Pr(C )
.20 4
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Conditional Probability –
Equally Likely Outcomes
Example: Conditional Probability
„
„Conditional
Probability in Case of Equally Likely
Outcomes
Pr( E | F ) =
[ number of outcomes in E ∩ F ]
[ number of outcomes in F ]
A sample of two balls are selected from an
urn containing 8 white balls and 2 green
balls. What is the probability that the
second ball selected is white given that the
first ball selected was white?
provided that [number of outcomes in F] ≠ 0.
Example: Conditional Probability
„The number of outcomes in "the first ball is
white" is 8⋅9 = 72. That is, the first ball must be
among the 8 white balls and the second ball can be
any of the 9 balls left.
„The number of outcomes in "the first ball is
white and the second ball is white" is 8⋅7 = 56.
Pr(2nd ball is white |1st ball is white) =
Product Rule
„
Product Rule If Pr(F) ≠ 0,
Pr(E ∩ F) = Pr(F) ⋅ Pr(E | F).
„ The product rule can be extended to three events.
Pr(E1 ∩ E2 ∩ E3) = Pr(E1) ⋅ Pr(E2 | E1) ⋅ Pr(E3| E1 ∩ E2)
56 7
=
72 9
Example: Product Rule
A sequence of two playing cards is drawn
at random (without replacement) from a
standard deck of 52 cards. What is the
probability that the first card is red and the
second is black?
Let
„ F = "the first card is red," and
„ E = "the second card is black."
„
Independent Events
¾
Events are said to be independent if the
probability of one event does not affect the
likelihood of occurrence of the other events(s).
¾
A collection of events is said to be independent
if for each collection of events chosen from
them, the probability that all the events occur
equals the product of the probabilities that each
occurs.
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Example: Independence
Independence
„
„Let
E and F be events. We say that E and F are
independent provided that
Pr(E ∩ F) = Pr(E) ⋅ Pr(F).
„Equivalently, they are independent provided that
Pr(E | F) = Pr(E) and Pr(F | E) = Pr(F).
Independence of a Set of Events
Let an experiment consist of observing the
results of drawing two consecutive cards
from a 52-card deck. Let E = "second card
is black" and F = "first card is red".
Are these two events independent?
Example: Independence of a Set
set of events is said to be independent if, for
each collection of events chosen from them, say E1,
E2, …, En, we have
Pr(E1 ∩ E2 ∩ … ∩ En) = Pr(E1) ⋅ Pr(E2) ⋅…⋅ Pr(En).
„A company manufactures stereo components.
Experience shows that defects in manufacture are
independent of one another. Quality control studies
reveal that
„2% of CD players are defective, 3% of amplifiers
are defective, and 7% of speakers are defective.
„A system consists of a CD player, an amplifier, and
2 speakers. What is the probability that the system is
not defective?
Example
Example
„A
„
The proportion of individuals in a certain
city earning more than $35,000 per year is
.25. The proportion of individuals earning
more than $35,000 and having a college
degree is .10. Suppose that a person is
randomly chosen and he turns out to be
earning more than $35,000. What is the
probability that he is a college graduate?
„
A stereo system contains 50 transistors. The
probability that a given transistor will fail in
100,000 hours of use is .0005. Assume that the
failures of the various transistors are
independent of one another. What is the
probability that no transistor will fail during the
first 100,000 hours of use?
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Example
Example
Let E and F be events with P(E) = .3, P(F) = .6,
and P(E ∪ F) = .7.
Find:
a.) P(E ∩ F)
Of the students at a certain college, 50% regularly
attend the football games, 30% are first-year students,
and 40% are upper-class students who do not regularly
attend football games. Suppose that a student is
selected at random.
b.) P(E | F)
a.) What is the probability that the person both is a firstyear student and regularly attends football games?
c.) P(F | E)
b.) What is the probability that the person regularly
attends football games given that he is a first-year
student?
„
„
d.) P(E′ ∩ F)
c.) What is the probability that the person is a first-year
student given that he regularly attends football games?
e.) P(E ′ | F)
Example
Example
„
Out of 250 students interviewed at a community
college, 90 were taking mathematics but not
computer science, 160 were taking mathematics,
and 50 were taking neither mathematics nor
computer science. Find the probability that a
student chosen at random was:
a.) taking just computer science.
b.) taking mathematics or computer science, but not
both.
c.) taking mathematics, given that the student was
taking computer science.
d.) taking computer science, given that the student
was not taking mathematics.
„
Two poker chips are selected at random
from an bag containing two white chips and
three red chips. What is the probability that
both chips are white given that at least one
of them is white?
Example
Find the probability that a
student selected at
random is:
a.) a senior.
b.) working full-time.
c.) working part-time, given
that the student is a firstyear student.
d.) a junior or senior, given
that the student does not
work.
Example
„
Works
full
time
Works
part
time
Firstyear
130
460
210
Soph.
100
500
150
Junior
80
420
100
Senior 200
300
50
Not
working
„
A bag contains eight purple marbles, six blue
marbles, and 12 red marbles. Four marbles are
selected from the bag.
a.) What is the probability that they are all purple?
b.) What is the probability that they are all the
same color?
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