Countingprinciple

CountingCounting
OutcomesOutcomes

Counting OutcomesCounting Outcomes
ObjectivesObjectives::
(1) To develop fluency with counting(1) To develop fluency with counting
strategies to determine the samplestrategies to determine the sample
space for an event.space for an event.
Essential QuestionsEssential Questions::
(1) How can I construct and use a(1) How can I construct and use a
frequency table (a.k.a. tree diagram)?frequency table (a.k.a. tree diagram)?
(2) How can I use the Fundamental(2) How can I use the Fundamental
Counting Principle to find the numberCounting Principle to find the number
of outcomes?of outcomes?

Have you ever seen or heard theHave you ever seen or heard the
Subway or Starbucks advertisingSubway or Starbucks advertising
campaigns where they talk about thecampaigns where they talk about the
10,000 different combinations of ways10,000 different combinations of ways
to order a sub or drink?to order a sub or drink?

Have you ever seen or heard theHave you ever seen or heard the
Subway or Starbucks advertisingSubway or Starbucks advertising
campaigns where they talk about thecampaigns where they talk about the
10,000 different combinations of ways10,000 different combinations of ways
to order a sub or drink?to order a sub or drink?
When companies like these makeWhen companies like these make
these claims they are using all thethese claims they are using all the
different condiments and ways todifferent condiments and ways to
serve a drink.serve a drink.

- These companies can use (2) ideas- These companies can use (2) ideas
related to combinations to make theserelated to combinations to make these
claims:claims:
(1) TREE DIAGRAMS(1) TREE DIAGRAMS
(2) THE FUNDAMENTAL(2) THE FUNDAMENTAL
COUNTING PRINCIPLECOUNTING PRINCIPLE

(1) TREE DIAGRAMS(1) TREE DIAGRAMS
A tree diagram is a diagram used to showA tree diagram is a diagram used to show
the total number of possible outcomes inthe total number of possible outcomes in
a probability experiment.a probability experiment.

(2) THE FUNDAMENTAL(2) THE FUNDAMENTAL
COUNTING PRINCIPLECOUNTING PRINCIPLE
The Fundamental Counting Principle usesThe Fundamental Counting Principle uses
multiplication of the number of ways eachmultiplication of the number of ways each
event in an experiment can occur to findevent in an experiment can occur to find
the number of possible outcomes in athe number of possible outcomes in a
sample space.sample space.

Example 1Example 1:: Tree Diagrams.Tree Diagrams.
A new polo shirt is released in 4 differentA new polo shirt is released in 4 different
colors and 5 different sizes. How manycolors and 5 different sizes. How many
different color and size combinationsdifferent color and size combinations
are available to the public?are available to the public?
Colors –Colors – (Red, Blue, Green, Yellow)(Red, Blue, Green, Yellow)
Styles –Styles – (S, M, L, XL, XXL)(S, M, L, XL, XXL)

Example 1Example 1:: Tree Diagrams.Tree Diagrams.
Answer.Answer.
RedRed BlueBlue Green YellowGreen Yellow
S M L XL XXLS M L XL XXL S M L XL XXLS M L XL XXL
S M L XL XXLS M L XL XXL S M L XL XXLS M L XL XXL
There areThere are 20 different combinations20 different combinations..

Example 1Example 1:: The Fundamental CountingThe Fundamental Counting
Principle.Principle.
A new polo shirt is released in 4 differentA new polo shirt is released in 4 different
colors and 5 different sizes. How manycolors and 5 different sizes. How many
different color and size combinationsdifferent color and size combinations
are available to the public?are available to the public?
Colors –Colors – (Red, Blue, Green, Yellow)(Red, Blue, Green, Yellow)
Styles –Styles – (S, M, L, XL, XXL)(S, M, L, XL, XXL)

Answer.Answer.
Number ofNumber of Number ofNumber of Number ofNumber of
Possible StylesPossible Styles Possible SizesPossible Sizes Possible Comb.Possible Comb.
44 xx 55 == 2020

 Tree Diagrams and The FundamentalTree Diagrams and The Fundamental
Counting Principle are two differentCounting Principle are two different
algorithms for finding sample space ofalgorithms for finding sample space of
a probability problem.a probability problem.
 However, tree diagrams work betterHowever, tree diagrams work better
for some problems and thefor some problems and the
fundamental counting principle worksfundamental counting principle works
better for other problems.better for other problems.

Example 2Example 2:: Tree Diagram.Tree Diagram.
Tamara spins a spinner twoTamara spins a spinner two
times. What is her probabilitytimes. What is her probability
of spinning a green on theof spinning a green on the
first spin and a blue on the second spin?first spin and a blue on the second spin?

Example 2Example 2:: Tree Diagram.Tree Diagram.
Tamara spins a spinner twoTamara spins a spinner two
times. What is her probabilitytimes. What is her probability
of spinning a green on theof spinning a green on the
first spin and a blue on the second spin?first spin and a blue on the second spin?
GreenGreen BlueBlue
GreenGreen BlueBlue GreenGreen BlueBlue
Only one outcome has green then blue, and there are 4Only one outcome has green then blue, and there are 4
possibilities…so the P(green, blue) = ¼ or .25 or 25%possibilities…so the P(green, blue) = ¼ or .25 or 25%

If a lottery game is made up of threeIf a lottery game is made up of three
digits from 0 to 9, what is thedigits from 0 to 9, what is the
probability of winning the game?probability of winning the game?

If a lottery game is made up of three digitsIf a lottery game is made up of three digits
from 0 to 9, what is the probability offrom 0 to 9, what is the probability of
winning if you buy 1 ticket?winning if you buy 1 ticket?
# of Possible# of Possible # of Possible# of Possible # of Possible# of Possible # of Possible# of Possible
DigitsDigits DigitsDigits DigitsDigits OutcomesOutcomes
10 x 10 x 10 =10 x 10 x 10 = 10001000
Because there are 1000 different possibilities, buying oneBecause there are 1000 different possibilities, buying one
ticket gives you a 1/1000 probability or 0.001 or 0.1% chanceticket gives you a 1/1000 probability or 0.001 or 0.1% chance
of winning.of winning.

Guided PracticeGuided Practice:: Determine the probabilityDetermine the probability
for each problem.for each problem.
(1) How many outfits are possible from a pair(1) How many outfits are possible from a pair
of jean or khaki shorts and a choice ofof jean or khaki shorts and a choice of
yellow, white, or blue shirt?yellow, white, or blue shirt?
(2) Scott has 5 shirts, 3 pairs of pants, and 4(2) Scott has 5 shirts, 3 pairs of pants, and 4
pairs of socks. How many different outfitspairs of socks. How many different outfits
can Scott choose with a shirt, pair ofcan Scott choose with a shirt, pair of
pants, and pair of socks?pants, and pair of socks?

Guided PracticeGuided Practice:: Determine the probabilityDetermine the probability
for each problem.for each problem.
(1)(1) Jean ShortsJean Shorts Khaki ShortsKhaki Shorts
YellowYellow White BlueWhite Blue Yellow White BlueYellow White Blue
JSYSJSYS11 JSWSJSWS22 JSBSJSBS33 KSYSKSYS44 KSWSKSWS55 KSBSKSBS66
(2) Number(2) Number NumberNumber NumberNumber NumberNumber
Of ShirtsOf Shirts Of PantsOf Pants Of SocksOf Socks Of OutfitsOf Outfits
5 x 3 x 4 =5 x 3 x 4 = 6060

Real World ExampleReal World Example:: The FundamentalThe Fundamental
Counting Principle.Counting Principle.
How many seven digit telephone numbersHow many seven digit telephone numbers
can be made up using the digits 0-9,can be made up using the digits 0-9,
without repetition?without repetition?

Real World ExampleReal World Example:: The FundamentalThe Fundamental
Counting Principle.Counting Principle.
How many seven digit telephone numbersHow many seven digit telephone numbers
can be made up using the digits 0-9,can be made up using the digits 0-9,
without repetition?without repetition?
Answer: 604,800 different numbersAnswer: 604,800 different numbers

Real World ExampleReal World Example:: Tree Diagram.Tree Diagram.
Kaitlyn tosses a coin 3 times. Draw aKaitlyn tosses a coin 3 times. Draw a
picture showing the possible outcomes.picture showing the possible outcomes.
What is the probability of getting atWhat is the probability of getting at
least 2 tails?least 2 tails?

Real World ExampleReal World Example:: Tree Diagram.Tree Diagram.
Kaitlyn tosses a coin 3 times. Draw aKaitlyn tosses a coin 3 times. Draw a
picture showing the possible outcomes.picture showing the possible outcomes.
What is the probability of getting atWhat is the probability of getting at
least 2 tails?least 2 tails?
Answer: P(at least 2 tails) = ½Answer: P(at least 2 tails) = ½

SummarySummary::
- A- A tree diagramtree diagram is used to show all of theis used to show all of the
possible outcomes, or sample space, in apossible outcomes, or sample space, in a
probability experiment.probability experiment.
- The- The fundamental counting principlefundamental counting principle cancan
be used to count the number of possiblebe used to count the number of possible
outcomes given an event that can happenoutcomes given an event that can happen
in some number of ways followed byin some number of ways followed by
another event that can happen in someanother event that can happen in some
number of different ways.number of different ways.

SummarySummary:: So when should I use a treeSo when should I use a tree
diagram or the fundamental countingdiagram or the fundamental counting
principle?principle?
- A- A tree diagramtree diagram is used to:is used to:
(1) show sample space;(1) show sample space;
(2) count the number of preferred outcomes.(2) count the number of preferred outcomes.
- The- The fundamental counting principlefundamental counting principle cancan
be used to:be used to:
(1) count the total number of outcomes.(1) count the total number of outcomes.

HomeworkHomework::
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Countingprinciple

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