Download Multiplication rule for independent events

CHAPTER 5
Probability: What Are
the Chances?
5.3
Conditional Probability
and Independence
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Conditional Probability and Independence
Learning Objectives
After this section, you should be able to:
 CALCULATE and INTERPRET conditional probabilities.
 USE the general multiplication rule to CALCULATE probabilities.
 USE tree diagrams to MODEL a chance process and CALCULATE
probabilities involving two or more events.
 DETERMINE if two events are independent.
 When appropriate, USE the multiplication rule for independent
events to COMPUTE probabilities.
The Practice of Statistics, 5th Edition
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What is Conditional Probability?
The probability we assign to an event can change if we know that some
other event has occurred. This idea is the key to many applications of
probability.
When we are trying to find the probability that one event will happen
under the condition that some other event is already known to have
occurred, we are trying to determine a conditional probability.
The probability that one event happens given that another event is
already known to have happened is called a conditional
probability.
Suppose we know that event A has happened. Then the
probability that event B happens given that event A has happened
is denoted by P(B | A).
Read | as “given that”
or “under the
condition that”
The Practice of Statistics, 5th Edition
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Calculating Conditional Probabilities
Calculating Conditional Probabilities
To find the conditional probability P(A | B), use the formula
P(A | B) =
P(A Ç B)
P(B)
The conditional probability P(B | A) is given by
P(B | A) =
The Practice of Statistics, 5th Edition
P(B Ç A)
P(A)
4
P(A | B) =
P(A Ç B)
P(B)
P(male and pierced ears)
P(male | pierced ears) 
P(pierced ears)
19
19
178
P(male | pierced ears) 

103 103
178
The Practice of Statistics, 5th Edition
5
Who Reads the Paper?
What is the probability that a randomly selected resident who reads USA Today
also reads the New York Times?
P( B | A) 
P( B  A)
P( A)
P(NYT and USA Today)
P(reads NYT | reads USA Today ) 
P(USAToday)
P( B | A) 
0.05
 0.125
0.40
There is a 12.5% chance that a randomly selected resident who reads
USA Today also reads the New York Times.
The Practice of Statistics, 5th Edition
6
CYU
will define We
events
has a landline
has a celland
phone.
willL:define
events L:and
hasC:
a landline
C: has a cell phone.
(a)
Cell phone NoCell
cell phone
phone No
Total
cell phone Total
Landline
0.51
0.09
0.600.09
Landline
0.51
0.60
No landline
0.38
0.02
0.400.02
No
landline
0.38
0.40
Total
0.89
0.11
1.000.11
Total
0.89
1.00
What(b)
isGthe probability that a randomly selected household withH
a
G
landline also has a cell phone?
H
P(cell phone and landline)
P(cell phone | landline) 
89
221
119 P(landline)
89
221
119
0.51 71
P(cell phone | landline) 
 0.85
0.60
71
To find the
probability
the household
has at user
least also
one of
twophone.
types of phones, we
There
is a 0.85 that
probability
that a landline
hasthe
a cell
(c) To find
thata the
household
at least
one of the two typ
ind the probability
that the
the probability
household has
landline,
a cellhas
phone,
or both.
to find
the+probability
has– a0.51
landline,
 ofC)
 the
= P(L)
P(C) – P(Lthat
C) =household
0.60 + 0.89
= 0.98a cell phone, or 7bot
TheP(L
Practice
Statistics,
5 Edition
th
Calculating Conditional Probabilities
Consider the two-way table on page 321. Define events
E: the grade comes from an EPS course, and
L: the grade is lower than a B.
Total
6300
1600
2100
Total 3392 2952
3656
10000
Find P(L)
P(L) = 3656 / 10000 = 0.3656
Find P(E | L)
P(E | L) = 800 / 3656 = 0.2188
Find P(L | E)
P(L| E) = 800 / 1600 = 0.5000
The Practice of Statistics, 5th Edition
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The General Multiplication Rule
General Multiplication Rule
The probability that events A and B both occur can be found
using the general multiplication rule
P(A ∩ B) = P(A) • P(B | A)
where P(B | A) is the conditional probability that event B occurs
given that event A has already occurred.
In words, this rule says that for both of two events to occur, first one
must occur, and then given that the first event has occurred, the
second must occur.
The Practice of Statistics, 5th Edition
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Teens With Online Profiles
• The Pew Internet and American Life Project find that 93% of
teenagers (ages 12 to 17) use the Internet, and that 55% of online
teens have posted a profile on a social-networking site.
Find the probability that a randomly selected teen uses the Internet
and has posted a profile. Show your work.
• There is about a 51% chance that a randomly selected teen uses
the Internet and has posted a profile on a social-networking site.
The Practice of Statistics, 5th Edition
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Tree Diagrams
The general multiplication rule is especially useful when a chance
process involves a sequence of outcomes. In such cases, we can use a
tree diagram to display the sample space.
Consider flipping a coin
twice.
What is the probability of
getting two heads?
Sample Space:
HH HT TH TT
So, P(two heads) = P(HH) = 1/4
The Practice of Statistics, 5th Edition
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Example: Tree Diagrams
The Pew Internet and American Life Project finds that 93% of
teenagers (ages 12 to 17) use the Internet, and that 55% of online
teens have posted a profile on a social-networking site.
What percent of teens are online and have posted a profile?
P(online) = 0.93
P(profile | online) = 0.55
P(online and have profile) = P(online)× P(profile | online)
= (0.93)(0.55)
= 0.5115
51.15% of teens are online and have posted
a profile.
The Practice of Statistics, 5th Edition
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• Tennis example on p.322
The Practice of Statistics, 5th Edition
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• YouTube example on p.324
(a) Draw a tree diagram to represent this situation.
(b) Find the probability that this person has visited a video-sharing site. Show
your work.
(c) Given that this person has visited a video-sharing site, find the probability
that he or she is aged 18 to 29. Show your work.
The Practice of Statistics, 5th Edition
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• CYU on p.326
The Practice of Statistics, 5th Edition
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Conditional Probability and Independence
When knowledge that one event has happened does not change the
likelihood that another event will happen, we say that the two events
are independent.
Two events A and B are independent if the occurrence of one
event does not change the probability that the other event will
happen. In other words, events A and B are independent if
P(A | B) = P(A) and P(B | A) = P(B).
When events A and B are independent, we can simplify the general
multiplication rule since P(B| A) = P(B).
Multiplication rule for independent events
If A and B are independent events, then the probability that A and
B both occur is
P(A ∩ B) = P(A) • P(B)
The Practice of Statistics, 5th Edition
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Testing for Independence
The events of interest are A: is male and B: has pierced ears.
For the two events to be independent, P( A | B )  P ( A) or P( B | A)  P ( B )
Since the conditional and unconditional probabilities are not the same the
two events are not independent.
The Practice of Statistics, 5th Edition
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• CYU on p.328
The Practice of Statistics, 5th Edition
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Multiplication Rule for Independent Events
Following the Space Shuttle Challenger disaster, it was determined that
the failure of O-ring joints in the shuttle’s booster rockets was to blame.
Under cold conditions, it was estimated that the probability that an
individual O-ring joint would function properly was 0.977.
Assuming O-ring joints succeed or fail independently, what is the
probability all six would function properly?
P( joint 1 OK and joint 2 OK and joint 3 OK and joint 4 OK and joint 5 OK
and joint 6 OK)
By the multiplication rule for independent events, this probability is:
P(joint 1 OK) · P(joint 2 OK) · P (joint 3 OK) • … · P (joint 6 OK)
= (0.977)(0.977)(0.977)(0.977)(0.977)(0.977) = 0.87
There’s an 87% chance that the shuttle would launch safely under similar
conditions (and a 13% chance that it wouldn’t).
The Practice of Statistics, 5th Edition
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• HIV testing example on p.329
– “At least 1” and “none” are opposites
• “Think about it” on p.331
• CYU on p.331
The Practice of Statistics, 5th Edition
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Conditional Probabilities and Independence
Section Summary
In this section, we learned how to…
 CALCULATE and INTERPRET conditional probabilities.
 USE the general multiplication rule to CALCULATE probabilities.
 USE tree diagrams to MODEL a chance process and CALCULATE
probabilities involving two or more events.
 DETERMINE if two events are independent.
 When appropriate, USE the multiplication rule for independent events
to COMPUTE probabilities.
The Practice of Statistics, 5th Edition
21