Download Section 6.5

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Conditional Probability
Conditional Probability of Equally Likely
Outcomes
Product Rule
Independence
Independence of a Set of Events
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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.
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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?
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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 ) .15 3
Pr( H | C ) 

 .
Pr(C )
.20 4
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Conditional
Outcomes
Probability in Case of Equally Likely
number of outcomes in E  F 

Pr( E | F ) 
 number of outcomes in F 
provided that [number of outcomes in F] ≠ 0.
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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?
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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(2
nd
56 7
ball is white |1 ball is white) 

72 9
st
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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)
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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."
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Pr(F)
= ½ since half the deck is red cards.
If we know the first card is red, then the
probability the second is black is
Therefore,
26
Pr( E | F )  .
51
1 26 13
Pr( E  F )  
 .
2 51 51
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Let
E and F be events. We say that E and F are
independent provided that
= Pr(E)  Pr(F).
Equivalently, they are independent provided that
Pr(E | F) = Pr(E) and Pr(F | E) = Pr(F).
Pr(E  F)
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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?
From the previous example, Pr(E | F) = 26/51.
Note that Pr(E) = 1/2.
Since they are not equal, E and F are not
independent.
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A
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).
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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?
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Let
C, A, S1, and S2 be events corresponding to
defective CD player, amplifier, speaker 1, and
speaker 2, respectively. Then
Pr(C) = .02, Pr(A) = .03, Pr(S1) = Pr(S2) = .07
Pr(C') = .98, Pr(A') = .97, Pr(S1') = Pr(S2') = .93
 S1‘ S2') = .98.97.932 = .822
Pr(C‘ A‘
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Pr(E|F), the conditional probability that E occurs
given that F occurs, is computed as
Pr(E F)/Pr(F). For a sample space with a finite
number of equally likely outcomes, it can be
computed as n(E F)/n(F).
 The product rule states that if Pr(F) ≠ 0, then
Pr(E F) = Pr(F) Pr(E|F).

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E and F are independent events if Pr(E F) 
= Pr(F)
Pr(E). Equivalently, E and F [with Pr(F) ≠ 0] are
independent events if Pr(E|F) = Pr(E).
 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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