Transcript 4.4 Notes

Section 4-4
Multiplication Rule:
Basics
Key Concept
The basic multiplication rule is used for finding
P(A and B), the probability that event A occurs
in a first trial and event B occurs in a second
trial.
If the outcome of the first event A somehow
affects the probability of the second event B, it
is important to adjust the probability of B to
reflect the occurrence of event A.
Notation
P(A and B) = P(event A occurs in a first trial and
event B occurs in a second trial)
Tree Diagrams
A tree diagram is a picture of the possible
outcomes of a procedure, shown as line
segments emanating from one starting point.
These diagrams are sometimes helpful in
determining the number of possible
outcomes in a sample space, if the number of
possibilities is not too large.
Tree Diagrams
This figure summarizes
the possible outcomes
for a true/false question
followed by a multiple
choice question.
Note that there are 10
possible combinations.
Example 1:
a.) Create a tree diagram for first tossing a coin
and then rolling a die.
Heads
Tails
Example 1:
b) How many possible combinations exist?
Heads
Tails
Example 1:
c) What is the probability of flipping Tails, and
then rolling a four?
Heads
Tails
Conditional Probability
Key Point
We must adjust the probability of the
second event to reflect the outcome of
the first event.
Conditional Probability
Important Principle
The probability for the second event B
should take into account the fact that the
first event A has already occurred.
Notation for Conditional Probability
P(B | A) represents the probability of event B
occurring after it is assumed that event A has
already occurred (read B | A as “B given A.”)
Dependent and Independent
Two events A and B are independent if the
occurrence of one does not affect the probability of
the occurrence of the other. (Several events are
similarly independent if the occurrence of any does
not affect the probabilities of the occurrence of the
others.) If A and B are not independent, they are said
to be dependent.
Dependent Events
Two events are dependent if the occurrence of
one of them affects the probability of the
occurrence of the other, but this does not
necessarily mean that one of the events is a
cause of the other.
Example 2: A bag contains 8 yellow marbles, 4 blue marbles
and 1 white marble. Billy found the probability of choosing a
blue marble, and then another blue marble. In doing so, Billy
replaced the first blue marble. Margo also found the probability
of choosing a blue marble, and then another blue marble. She
did not replace the first blue marble. Which situation is a
dependent event?
Treating Dependent Events as Independent
Some calculations are cumbersome, but they can
be made manageable by using the common
practice of treating events as independent when
small samples are drawn from large populations.
In such cases, it is rare to select the same item
twice.
The 5% Guideline for Cumbersome
Calculations
If a sample size is no more than 5% of the
size of the population, treat the selections
as being independent (even if the
selections are made without replacement,
so they are technically dependent).
Example 3: For the given pair of events, classify them
as independent or dependent. If the two events are
technically dependent but can be treated as if they
are independent according to the 5% guideline,
consider them to be independent.
Randomly selecting a TV viewer who is watching
Saturday Night Live.
Randomly selecting a second TV viewer who is
watching Saturday Night Live.
Example 4: For the given pair of events, classify them as
independent or dependent. If the two events are
technically dependent but can be treated as if they are
independent according to the 5% guideline, consider them
to be independent.
Wearing plaid shorts with black socks and sandals.
Asking someone on a date and getting a positive response.
Formal Multiplication Rule
P(A and B) = P(A) • P(B | A)
*Note that if A and B are independent events,
P(B | A) is really the same as P(B).
Intuitive Multiplication Rule
When finding the probability that event A occurs in
one trial and event B occurs in the next trial,
multiply the probability of event A by the probability
of event B, but be sure that the probability of event
B takes into account the previous occurrence of
event A.
Applying the Multiplication Rule
Be sure to find the probability of
event B assuming that event A has
already occurred.
Caution
When applying the multiplication rule,
always consider whether the events are
independent or dependent, and adjust the
calculations accordingly.
Example 5: Use the data in the following table, which
summarizes blood groups and Rh types for 100 subjects. These
values may vary in different regions according to the ethnicity
of the population.
Group
Type
O
A
B
AB
Rh+
39
35
8
4
Rh–
6
5
2
1
If 2 of the 100 subjects are randomly selected, find the
probability that they are both group O and type Rh+.
a) Assume that the selections are made with replacement.
Example 5: Use the data in the following table, which
summarizes blood groups and Rh types for 100 subjects.
These values may vary in different regions according to the
ethnicity of the population.
Group
Type
O
A
B
AB
Rh+
39
35
8
4
Rh–
6
5
2
1
If 2 of the 100 subjects are randomly selected, find the
probability that they are both group O and type Rh+.
b) Assume that the selections are made without replacement.
Multiplication Rule for Several
Events
In general, the probability of any sequence of
independent events is simply the product of
their corresponding probabilities.
Example 6: Use the data in the following table, which summarizes blood groups and
Rh types for 100 subjects. These values may vary in different regions according to
the ethnicity of the population.
Group
Type
O
A
B
AB
Rh+
39
35
8
4
Rh–
6
5
2
1
People with blood that is group O and type Rh– are considered to be universal
donors, because they can give blood to anyone. If 4 of the 100 subjects are
randomly selected, find the probability that they are all universal donors.
a) Assume that the selections are made with replacement.
Example 6: Use the data in the following table, which summarizes blood groups and
Rh types for 100 subjects. These values may vary in different regions according to
the ethnicity of the population.
Group
Type
O
A
B
AB
Rh+
39
35
8
4
Rh–
6
5
2
1
People with blood that is group O and type Rh– are considered to be universal
donors, because they can give blood to anyone. If 4 of the 100 subjects are
randomly selected, find the probability that they are all universal donors.
b) Assume that the selections are made without replacement.
Principle of Redundancy
One design feature contributing to reliability is the
use of redundancy, whereby critical components
are duplicated so that if one fails, the other will
work. For example, single-engine aircraft now have
two independent electrical systems so that if one
electrical system fails, the other can continue to
work so that the engine does not fail.
Summary of Fundamentals
*In the addition rule, the word “or” in P(A or B)
suggests addition. Add P(A) and P(B), being
careful to add in such a way that every outcome
is counted only once.
*In the multiplication rule, the word “and” in P(A
and B) suggests multiplication. Multiply P(A) and
P(B), but be sure that the probability of event B
takes into account the previous occurrence of
event A.
Example 7: A quick quiz consists of a true/false question followed by a
multiple-choice question with four possible answers (a, b, c, d). An
unprepared student makes random guesses for both answers.
a) Consider the event of being correct with the first guess and the event
of being correct with the second guess. Are those two events
independent?
b) What is the probability that both answers are correct?
Example 7: A quick quiz consists of a true/false question
followed by a multiple-choice question with four possible
answers (a, b, c, d). An unprepared student makes random
guesses for both answers.
c) Based on the results, does guessing appear to be a good
strategy?
Example 8: The Wheeling Tire Company produced a batch of
5000 tires that includes exactly 200 that are defective.
a) If 4 tires are randomly selected for installation on a car, what
is the probability that they are all good?
Example 8: The Wheeling Tire Company produced a batch of 5000
tires that includes exactly 200 that are defective.
b) If 100 tires are randomly selected for shipment to an outlet, what
is the probability that they are all good? Should this outlet plan to
deal with defective tires returned by consumers?
Example 9: The principle of redundancy is used when system
reliability is improved through redundant or backup components.
Assume that your alarm clock has a 0.9 probability of working on
any given morning.
a) What is the probability that your alarm clock will not work on the
morning of an important exam?
b) If you have two such alarm clocks, what is the probability that
they both fail on the morning of an important exam?
Example 9 continued: The principle of redundancy is used when
system reliability is improved through redundant or backup
components. Assume that your alarm clock has a 0.9 probability of
working on any given morning.
c) With one alarm clock, you have a 0.9 probability of being
awakened. What is the probability of being awakened if you have
two alarm clocks?
Example 9 continued: The principle of redundancy is
used when system reliability is improved through
redundant or backup components. Assume that your
alarm clock has a 0.9 probability of working on any
given morning.
d) Does a second alarm clock result in greatly
improved reliability?