Transcript Notes on Tree Diagrams and Multiplication Rule

Simple Sample
Spaces…Tree
Diagrams
Warm Up – Describe the
Correlation
x = Weight of luggage
y = Cost to fly your luggage on an
airplane.
Strong Positive
Warm Up – Describe the
Correlation
x = Number of Chiropractic
Adjustments
y = Amount of Back Pain
Strong Negative
Warm Up – What Kind of
Sampling?
Every 5th adult entering an airport is
checked for extra security screening.
Systematic
Warm Up – What Kind of
Sampling?
A writer for an art magazine randomly
selects and interviews 50 male and 50
female artists.
Stratified
Warm Up
A student scores a 75 on a geography test with
a mean of 80 and a standard deviation of 5.
The same student scores a 249 on his math
test that had a mean of 300 and a standard
deviation of 34.
On which test did the student score better
relative to the other students in the class?
The student
scored better on
geography test.
Objective
Find
Sample Space
using Tree Diagrams
and the Multiplication
Rule.
Relevance
 Learn
various methods of
finding out how many
possible outcomes of a
probability experiment are
possible.
 Use this information to find
probability.
Definitions


Outcome – a
particular result of
an experiment
Event – any subset
of all outcomes.
(This is called the
Sample Space)

Sample Space:
The set of all possible
outcomes. Outcomes
cannot overlap. All
outcomes must be
represented. Can find by:
1. A List
2. A Tree Diagram
3. Grid
Example

List the sample space for
drawing one card from a
deck of cards.
Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A
Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A
Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A
Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, J, Q, K, A
52 Outcomes
Example……

Find the sample
space for rolling 2
dice.
Die 2
Die 1
1
2
3
4
5
6
1
1,1
1,2
1,3
1,4
1,5
1,6
2
2,1
2,2
2,3
2,4
2,5
2,6
3
3,1
3,2
3,3
3,4
3,5
3,6
4
4,1
4,2
4,3
4,4
4,5
4,6
5
5,1
5,2
5,3
5,4
5,5
5,6
6
6,1
6,2
6,3
6,4
6,5
6,6
Example

Find the sample space for the gender of
children if a family has 3 children.

See if you can list all of the possibilities.
BBB,BBG,BGB,BGG,GBB,GBG,GGB,GGG
8 Outcomes
Tree Diagrams

Tree Diagram – a device used to list all
possibilities in a systematic way.

Each branch lists the choices.

Let’s do a tree diagram of the previous example
with the gender of 3 children. Remember, we
had 8 outcomes.
Tree Diagram of 3 Children
Gender
B
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B
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G

B

B

G

G
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
Start
B

B

G
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G
B
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G
G
Example

Display the sample space using a tree
diagram if a coin is tossed and a die is
rolled.
Coin Tossed & Die Rolled - 12 Outcomes:
H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6
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H
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1
2
3
4
5
6
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Start
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T
1
2
3
4
5
6
You Try

Draw a tree diagram if you toss one coin
and spin a 4 section spinner.
Answer……8 Outcomes:
H1,H2,H3,H4,T1,T2,T3,T4
1


2

H

3
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4
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1
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Start
2
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3
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T
4
Example
A lunch menu has a choice of the following
(one from each category)
Meat: Hamburger, Hot Dog, Chicken
Starch: French Fries, Baked Potato
Drink: Coke, Tea
 List all of the possible combinations.

Answer……12 Outcomes

Coke

FF

Tea


Hamburger
Coke

BP

Tea
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
Coke

FF

Tea
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
Hot Dog
Coke

BP

Tea

Coke

FF

Tea
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
Chicken
Coke
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

BP
Tea
Example

Sue and Tom play in a tournament. The
1st person to win 2 out of 3 games is the
winner. Find all the possible outcomes.
Answer……6 Outcomes
Game Over
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Sue


Sue
Sue

Tom

Tom
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
Sue

Sue

Tom
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
Tom
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
Tom
Game Over
Multiplication Rule…
Section 4.3
Independent / Dependent
Definition

Two events E and F are independent if the
occurrence of event E in a probability
experiment does not affect the probability of
event F.

Two events are dependent if the occurrence of
event E in a probability experiment affects the
probability of event F.
Examples
Independent
Draw a queen from a deck
of cards.
Replace it.
Draw another queen.
Roll a die and get a 6.
Roll a 2nd die and get a 3.
Dependent
Draw a card from a deck.
Do NOT replace it.
Draw another card.
Eating too many calories.
Putting on Weight.
Situation
Event 1: Draw a card.
Independent
or
Dependent?
Independent
Event 2: Roll a die.
Event 1: Earned a
Bachelor’s Degree
Event 2: Earn more than
$100,000 per year.
Dependent
Situation
Event 1: Being Female
Independent
or
Dependent?
Independent
Event 2: Having Brown
Hair
Event 1: Complete your
Homework
Event 2: Making a 100 on
the HW grade
Dependent
Situation
Event 1: Having your car
start
Independent
or
Dependent?
Dependent
Event 2: Making it to
statistics class on time
Event 1: Play the
California Lottery
Event 2: Play the South
Carolina Lottery
Independent
Multiplication Counting Rule
 When
2 events are independent,
multiply the # of possibilities in
each category to get the total # of
possibilities.
Example

Find the number of items in the sample
space of a license plate containing 3
letters and 3 numbers (digits) that can
be repeated.
26  26  26 10 10 10  17,576,000
Example

Find the number of items in the sample
space of a license plate containing 3
letters and 3 numbers (digits) that can
NOT be repeated.
26  25  24 10  9  8  11232000
Example
 How
many ways can a nurse visit
4 places.
4  3  2 1  24
Example

Find the number of items in the sample
space of a 5 digit I.D. tag if the numbers
a. Can be repeated
10 10 10 10 10  100000
b. Can NOT be repeated.
10  9  8  7  6  30240
Example
 How
many ways can someone
read 5 books?
5  4  3  2 1  120
Example
How many outcomes are in the sample
space if a person selects 3 items from a
jar with 6 items
 A. With Replacement

6  6  6  216

B. Without Replacement
6  5  4  120