Download Simple Sample Spaces…Tree Diagrams

Simple Sample
Spaces…Tree
Diagrams
Example
Outcome – a particular
result of an experiment
Sample Points –
Individual outcomes of
the sample space.
Event – any subset of 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. Lattice Grid
Example……
List the sample space for
drawing one card from a
deck of cards.
Answer. 52 Outcomes
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
Find the sample
space for rolling 2
dice.
Die 2
Die 1
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.
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
Answer……
BBB,BBG,BGB,GBB,GGG,GGB,GBG,BGG
8 Outcomes
1
Tree Diagrams……
Tree Diagram – a device used to list all
possibilities in a systematic way.
Tree Diagram of 3 Children
Gender
B
B
G
B
B
G
Each branch lists the choices.
G
Start
B
B
Let’s do a tree diagram of the previous example
with the gender of 3 children. Remember, we
had 8 outcomes.
Example……
Display the sample space using a tree
diagram if a coin is tossed and a die is
rolled.
G
G
B
G
G
Coin Tossed & Die Rolled - 12 Outcomes:
H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6
H
1
2
3
4
5
6
Start
T
You Try……
Draw a tree diagram if you toss one coin
and spin a 4 section spinner.
1
2
3
4
5
6
Answer……8 Outcomes:
H1,H2,H3,H4,T1,T2,T3,T4
1
2
H
3
4
1
Start
2
3
T
4
2
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
Coke
FF
Tea
Hot Dog
Coke
BP
Tea
Coke
FF
Tea
Chicken
Coke
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
Sue
Sue
Sue
Tom
Tom
Sue
Sue
Tom
Tom
Tom
Game Over
Independent Event……
Multiplication Rule…
Section 4.3
Two events A and B are
independent events if
the fact that A occurs
does NOT affect the
probability of B occurring.
Example:
a. Roll a die and get a 6;
then roll a 2nd die and
get a 3.
b. Draw a queen
from a deck of
cards, replace it and
draw another queen.
3
Identify the following as independent or
dependent events……
1. Being a female
and having brown
hair
1. Independent
2. Completing the
homework and
getting a 100 for
a HW grade
2. Dependent
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.
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.
Multiplication Rule……
When 2 events are independent, multiply
the # of possibilities in each category to
get the total # of possibilities.
Answer……
26 ⋅ 26 ⋅ 26 ⋅ 10 ⋅ 10 ⋅ 10 = 17,576,000
Answer…..
26 ⋅ 25 ⋅ 24 ⋅10 ⋅ 9 ⋅ 8 = 11232000
4
Example……
Answer……24
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
Answer…….
Can Be Repeated:
10 ⋅10 ⋅10 ⋅10 ⋅10 = 100000
b. Can NOT be repeated.
Answer……
Can NOT be repeated:
Example……
How many ways can someone read 5
books?
10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 = 30240
5
Answer……
Example……
How many outcomes are in the sample
space if a person selects 3 items from a
jar with 6 items
A. With Replacement
B. Without Replacement
5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 120
Answer…..
With Replacement:
6 ⋅ 6 ⋅ 6 = 216
Answer……
Without Replacement:
6 ⋅ 5 ⋅ 4 = 120
6