Download (1) TREE DIAGRAMS

Counting
Outcomes
Counting Outcomes
Objectives:
(1) To develop fluency with counting
strategies to determine the sample
space for an event.
Essential Questions:
(1) How can I construct and use a
frequency table (a.k.a. tree diagram)?
(2) How can I use the Fundamental
Counting Principle to find the number
of outcomes?
Counting Outcomes
Have you ever seen or heard the
Subway
or
Starbucks
advertising
campaigns where they talk about the
10,000 different combinations of ways
to order a sub or drink?
Counting Outcomes
Have you ever seen or heard the
Subway
or
Starbucks
advertising
campaigns where they talk about the
10,000 different combinations of ways
to order a sub or drink?
When companies like these make
these claims they are using all the
different condiments and ways to
serve a drink.
Counting Outcomes
- These companies can use (2) ideas
related to combinations to make these
claims:
(1) TREE DIAGRAMS
(2) THE FUNDAMENTAL
COUNTING PRINCIPLE
Counting Outcomes
(1) TREE DIAGRAMS
A tree diagram is a diagram used to show
the total number of possible outcomes in
a probability experiment.
Counting Outcomes
(2) THE FUNDAMENTAL
COUNTING PRINCIPLE
The Fundamental Counting Principle uses
multiplication of the number of ways each
event in an experiment can occur to find
the number of possible outcomes in a
sample space.
Counting Outcomes
Example 1: Tree Diagrams.
A new polo shirt is released in 4 different
colors and 5 different sizes. How many
different color and size combinations are
available to the public?
Colors – (Red, Blue, Green, Yellow)
Sizes – (S, M, L, XL, XXL)
Counting Outcomes
Example 1: Tree Diagrams.
Answer.
Red
Blue
S M L XL XXL
Green
Yellow
S M L XL XXL
S M L XL XXL
S M L XL XXL
There are 20 different combinations.
Counting Outcomes
Example 1: The Fundamental Counting
Principle.
A new polo shirt is released in 4 different
colors and 5 different sizes. How many
different color and size combinations are
available to the public?
Colors – (Red, Blue, Green, Yellow)
Sizes – (S, M, L, XL, XXL)
Counting Outcomes
Example 1: The Fundamental Counting
Principle.
Answer.
Number of
Possible Colors
4
Number of
Possible Sizes
x
5
Number of
Possible Comb.
=
20
Counting Outcomes
 Tree
Diagrams and The Fundamental
Counting Principle are two different
algorithms for finding sample space of
a probability problem.
 However,
tree diagrams work better
for some problems and the
fundamental counting principle works
better for other problems.
Counting Outcomes
Example 2: Tree Diagram.
Tamara spins a spinner two
times. What is her probability
of spinning a green on the
first spin and a blue on the second spin?
Counting Outcomes
Example 2: Tree Diagram.
Tamara spins a spinner two
times. What is her probability
of spinning a green on the
first spin and a blue on the second spin?
Green
Blue
Green
Blue
Green
Blue
Only one outcome has green then blue, and there are 4
possibilities…so the P(green, blue) = ¼ or .25 or 25%
Counting Outcomes
Example 3: The Fundamental Counting
Principle.
If a lottery game is made up of three
digits from 0 to 9, what is the
probability of winning the game?
Counting Outcomes
Example 3: The Fundamental Counting
Principle.
If a lottery game is made up of three digits
from 0 to 9, what is the probability of
winning if you buy 1 ticket?
# of Possible
Digits
10
x
# of Possible
Digits
10
# of Possible
Digits
x
10
# of Possible
Outcomes
=
1000
Because there are 1000 different possibilities, buying one
ticket gives you a 1/1000 probability or 0.001 or 0.1%
chance of winning.
Counting Outcomes
Guided Practice: Determine the probability
for each problem.
(1) How many outfits are possible from a pair
of jean or khaki shorts and a choice of
yellow, white, or blue shirt?
(2) Scott has 5 shirts, 3 pairs of pants, and 4
pairs of socks. How many different outfits
can Scott choose with a shirt, pair of
pants, and pair of socks?
Counting Outcomes
Guided Practice: Determine the probability
for each problem.
(1)
Jean Shorts
Yellow
JSYS1
White
JSWS2
(2) Number
Of Shirts
5
Khaki Shorts
Blue
JSBS3
Number
Of Pants
x
3
Yellow
KSYS4
White
KSWS5
Number
Of Socks
x
4
Blue
KSBS6
Number
Of Outfits
=
60
Counting Outcomes
Real World Example: The Fundamental
Counting Principle.
How many seven digit telephone numbers
can be made up using the digits 0-9,
without repetition?
Counting Outcomes
Real World Example: The Fundamental
Counting Principle.
How many seven digit telephone numbers
can be made up using the digits 0-9,
without repetition?
Answer: 604,800 different numbers
Counting Outcomes
Real World Example: Tree Diagram.
Kaitlyn tosses a coin 3 times. Draw a
picture showing the possible outcomes.
What is the probability of getting at
least 2 tails?
Counting Outcomes
Real World Example: Tree Diagram.
Kaitlyn tosses a coin 3 times. Draw a
picture showing the possible outcomes.
What is the probability of getting at
least 2 tails?
Answer: P(at least 2 tails) = ½
Counting Outcomes
Summary:
- A tree diagram is used to show all of the
possible outcomes, or sample space, in a
probability experiment.
- The fundamental counting principle can
be used to count the number of possible
outcomes given an event that can happen
in some number of ways followed by
another event that can happen in some
number of different ways.
Counting Outcomes
Summary: So when should I use a tree
diagram or the fundamental counting
principle?
- A tree diagram is used to:
(1) show sample space;
(2) count the number of preferred outcomes.
- The fundamental counting principle can
be used to:
(1) count the total number of outcomes.