and Tree Diagrams 4 Use the general multiplication rule to calculate probabilities Use a tree diagram to model a chance process involving a sequence of outcomes Calculate conditional probabilities using tree diagrams ID: 934747
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Slide1
Probability
Lesson 4.5The General Multiplication Rule and Tree Diagrams
4
Slide2Use the general multiplication rule to calculate probabilities.
Use a tree diagram to model a chance process involving a sequence of outcomes.Calculate conditional probabilities using tree diagrams.
The General Multiplication Rule and Tree Diagrams
Slide3The General Multiplication Rule and Tree Diagrams
Suppose that A and B are two events resulting from the same chance process. We can find the probability
P
(A or B) with the general addition rule:
P(A or B
) =
P
(A) + P(B) − P(A and B)How do we find the probability that both events happen, P(A and B)?
General Multiplication Rule
For any chance process, the probability that events A and B both occur can be
found using
the
general multiplication rule
:
P
(A and B) =
P
(
A)
•
P
(B
| A
)
Slide4The General Multiplication Rule and Tree Diagrams
The general multiplication rule says that for both of two events to occur, first one must occur. Then, given that the
first
event has occurred, the second must occur. To
confirm that this result is correct, start with the conditional probability formula
The numerator gives the probability we want because
P
(B and A) is the same as
P
(A and B). Multiply both sides of the above equation by
P
(A) to get
Slide5Was there divine intervention for Tebow
?The general multiplication rule
PROBLEM:
Tim
Tebow is an American football player who enjoyed tremendous success in the 2011-2012 season. In 2012,
Sporting News
reported that 70% of U.S. adults had heard of Tim
Tebow and that 43% of those who had heard of him believed that “divine intervention was at least partly responsible for his success.” Suppose that these results are accurate. Find the probability that a randomly selected U.S. adult had heard of Tim Tebow and believed that divine intervention was at least partly responsible for his success. Show your work.P(heard of Tebow and believes divine intervention)
= P(heard of Tebow
) · P(divine intervention | heard of Tebow)= (0.70)(0.43) = 0.301
Slide6The General Multiplication Rule and Tree Diagrams
Shannon hits the snooze button on her alarm on 60% of school days. If she hits snooze, there is a 0.70 probability that she makes it to her first
class on time. If she doesn’t hit snooze and gets up right away, there is a 0.90 probability that she makes it to class on time. Suppose we select a school day at random and record whether
Shannon
hits the snooze button and whether she arrives in class on time.
We can use a
tree
diagram to show this chance process.
Slide7The General Multiplication Rule and Tree Diagrams
Tree Diagram
A
tree diagram
shows the sample space of a chance process involving multiple stages. The probability of each outcome is shown on the corresponding branch of the tree. All probabilities after the first stage are conditional probabilities.
What is the probability that Shannon hits the snooze button and is late for class on a randomly selected school day?
The
general multiplication rule provides the answer:
P
(hits snooze and late)
=
P
(hits snooze)
• P(late | hits snooze) = (0.60)(0.30) = 0.18The previous calculation amounts to multiplying probabilities along the branches of the tree diagram.
Slide8The General Multiplication Rule and Tree Diagrams
What’s the probability that Shannon is late to class on a randomly selected school day? There are two ways this can happen: Shannon can hit the snooze button and be late or she cannot hit snooze and be late. Because these outcomes are mutually exclusive,
P
(late) =
P
(hits snooze and late) +
P
(doesn’t hit snooze and late)
Slide9Can you judge an e-book by its e-cover?
Tree diagramsPROBLEM:
Recently, Harris Interactive reported that 20% of Millennials (ages 18-36), 25% of Gen Xers (ages 37-48), 21% of Baby Boomers (ages 49-67), and 17% of Matures (ages 68 and older) read more e-books than paper books. According to the U.S. Census Bureau, 34% of those 18 and over are Millennials, 22% are Gen Xers, 30% are Baby boomers, and 14% are Matures. Assume that all of these results are accurate. Suppose that we select one U.S. adult at random.
(a) Draw a tree diagram to model this chance process.
(b) Find the probability that a randomly chosen person reads more e-books than paper books.
Slide10Can you judge an e-book by its e-cover?
Tree diagrams(
a) Draw a tree diagram to model this chance process.
Slide11Can you judge an e-book by its e-cover?
Tree diagrams(
b) Find the probability that a randomly chosen person reads more e-books than paper books.
P
(reads
more e-books)
=
(0.34)(0.20) + (0.22)(0.25) + (0.30)(0.21) + (0.14)(0.17) = 0.0680 + 0.0550 + 0.0630 + 0.0238= 0.2098
Slide12The General Multiplication Rule and Tree Diagrams
Some interesting conditional probability questions involve “going in reverse” on a tree diagram. For instance, suppose that Shannon is late for class on a randomly chosen school day. What is the probability that she hit the snooze button that
morning
?
We
can use the information from the tree diagram and the conditional probability formula to do the required calculation
:
P(late) = P(hits snooze and late) + P(doesn’t hit snooze and late)
Slide13Who reads e-books?
Tree diagrams and conditional probability
PROBLEM:
Recently, Harris Interactive reported that 20% of Millennials (ages 18-36), 25% of Gen Xers (ages 37-48), 21% of Baby Boomers (ages 49-67), and 17% of Matures (ages 68 and older) read more e-books than paper books. According to the U.S. Census Bureau, 34% of those 18 and over are Millennials, 22% are Gen Xers, 30% are Baby boomers, and 14% are Matures. Assume that all of these results are accurate. A randomly selected U.S.
adult
reads more
e-books than hard cover books. What is the probability that this person is a Millennial
?
Slide14LESSON APP 4.5
Lactose intolerance causes difficulty in
digesting dairy
products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States (not
including other
groups and people who consider themselves to belong to more than one race), 82% of the population is white, 14% is black, and 4% is Asian. Moreover, 15% of whites, 70% of blacks, and 90% of Asians are lactose
intolerant.
Suppose
we select a U.S. person at random.
Not Milk?
Construct a tree
diagram
to represent this situation
.
Find
the probability that the person is lactose intolerant
.
Given
that the chosen person is lactose intolerant, what is the probability that he or she is Asian?
Slide15LESSON APP 4.5
Not Milk?
Construct a tree
diagram
to represent this situation
.
Find
the probability that the person is lactose intolerant
.
Given
that the chosen person is lactose intolerant, what is the probability that he or she is Asian?
Slide16Use the general multiplication rule to calculate probabilities.
Use a tree diagram to model a chance process involving a sequence of outcomes.Calculate conditional probabilities using tree diagrams.
The General Multiplication Rule and Tree Diagrams