Counting Outcomes

CountingCounting

OutcomesOutcomes

Counting OutcomesCounting Outcomes

ObjectivesObjectives::(1) To develop fluency with (1) To develop fluency with

counting strategies to determine counting strategies to determine the sample space for an event.the sample 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 frequency table (a.k.a. tree diagram)?diagram)?

(2) How can I use the Fundamental (2) How can I use the Fundamental Counting Principle to find the Counting Principle to find the number of outcomes?number of outcomes?


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


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

When companies like these When companies like these make these claims they are make these claims they are using all the different using all the different condiments and ways to serve condiments and ways to serve a drink.a drink.


- These companies can use (2) - These companies can use (2) ideas related to combinations to ideas related to combinations to make these claims:make these claims:

(1) TREE DIAGRAMS(1) TREE DIAGRAMS

(2) THE FUNDAMENTAL (2) THE FUNDAMENTAL COUNTING COUNTING

PRINCIPLEPRINCIPLE


(1) TREE DIAGRAMS(1) TREE DIAGRAMS

A tree diagram is a diagram used to A tree diagram is a diagram used to show the total number of possible show the total number of possible outcomes in a probability outcomes in a probability experiment.experiment.


(2) THE FUNDAMENTAL (2) THE FUNDAMENTAL COUNTING PRINCIPLECOUNTING PRINCIPLE

The Fundamental Counting Principle The Fundamental Counting Principle uses multiplication of the number of uses multiplication of the number of ways each event in an experiment ways each event in an experiment can occur to find the number of can occur to find the number of possible outcomes in a sample possible outcomes in a sample space.space.


Example 1Example 1:: Tree Diagrams.Tree Diagrams.

A new polo shirt is released in 4 A new polo shirt is released in 4 different colors and 5 different sizes. different colors and 5 different sizes. How many different color and size How many different color and size combinations are available to the combinations are available to the public?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 Blue Blue Green Yellow Green Yellow

S M L XL XXLS M L XL XXL S M L XL XXL S M L XL XXL

S M L XL XXLS M L XL XXL S M L XL XXL S M L XL XXL

There are There are 20 different combinations20 different combinations..


Example 1Example 1:: The Fundamental The Fundamental Counting Principle.Counting Principle.

A new polo shirt is released in 4 A new polo shirt is released in 4 different colors and 5 different sizes. different colors and 5 different sizes. How many different color and size How many different color and size combinations are available to the combinations are available to the public?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 of Number of Number of Number of

Possible StylesPossible Styles Possible SizesPossible Sizes Possible Possible Comb.Comb.

44 x x 5 5 = =2020


Tree Diagrams and The Tree Diagrams and The Fundamental Counting Principle Fundamental Counting Principle are two different algorithms for are two different algorithms for finding sample space of a finding sample space of a probability problem.probability problem.

However, tree diagrams work However, tree diagrams work better for some problems and the better for some problems and the fundamental counting principle fundamental counting principle works better for other problems.works 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 first spin and a blue on the second spin?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 Blue Blue

GreenGreen BlueBlue Green Green Blue Blue

Only one outcome has green then blue, and there are Only one outcome has green then blue, and there are 4 possibilities…so the P(green, blue) = ¼ or .25 or 4 possibilities…so the P(green, blue) = ¼ or .25 or 25%25%



If a lottery game is made up of three If a lottery game is made up of three digits from 0 to 9, what is the digits 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 digits If a lottery game is made up of three digits from 0 to 9, what is the probability of from 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 Possibleof Possible

DigitsDigits Digits Digits Digits Digits Outcomes Outcomes

10 x 10 x 10 = 10 x 10 x 10 = 10001000

Because there are 1000 different possibilities, buying one Because there are 1000 different possibilities, buying one ticket gives you a 1/1000 probability or 0.001 or 0.1% ticket gives you a 1/1000 probability or 0.001 or 0.1% chance of winning.chance of winning.


Guided PracticeGuided Practice:: Determine the Determine the probability for each problem.probability for each problem.

(1) How many outfits are possible from a (1) How many outfits are possible from a pair of jean or khaki shorts and a choice pair of jean or khaki shorts and a choice of yellow, white, or blue shirt?of yellow, white, or blue shirt?

(2) Scott has 5 shirts, 3 pairs of pants, (2) Scott has 5 shirts, 3 pairs of pants, and 4 pairs of socks. How many and 4 pairs of socks. How many different outfits can Scott choose with a different outfits can Scott choose with a shirt, pair of pants, and pair of socks?shirt, pair of pants, and pair of socks?


Guided PracticeGuided Practice:: Determine the Determine the probability for each problem.probability for each problem.

(1) (1) Jean Shorts Jean Shorts Khaki Shorts Khaki Shorts

Yellow Yellow White Blue White Blue Yellow White Blue Yellow White Blue

JSYSJSYS11 JSWS JSWS22 JSBS JSBS33 KSYS KSYS44 KSWS KSWS55 KSBS KSBS66

(2) Number(2) Number Number Number NumberNumber Number Number

Of ShirtsOf Shirts Of PantsOf Pants Of SocksOf Socks Of Outfits Of Outfits

5 x 3 x 4 = 5 x 3 x 4 = 6060


Real World ExampleReal World Example:: The The Fundamental Counting Principle.Fundamental Counting Principle.

How many seven digit telephone How many seven digit telephone numbers can be made up using the numbers can be made up using the digits 0-9, without repetition?digits 0-9, without repetition?


Real World ExampleReal World Example:: The The Fundamental Counting Principle.Fundamental Counting Principle.

How many seven digit telephone How many seven digit telephone numbers can be made up using the numbers can be made up using the digits 0-9, without repetition?digits 0-9, 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 a Kaitlyn tosses a coin 3 times. Draw a picture showing the possible picture showing the possible outcomes. What is the probability of outcomes. What is the probability of getting at least 2 tails?getting at least 2 tails?


Real World ExampleReal World Example:: Tree Diagram.Tree Diagram.

Kaitlyn tosses a coin 3 times. Draw a Kaitlyn tosses a coin 3 times. Draw a picture showing the possible picture showing the possible outcomes. What is the probability of outcomes. What is the probability of getting at least 2 tails?getting at 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 is used to show all of the possible outcomes, or sample the possible outcomes, or sample space, in a probability experiment.space, in a probability experiment.

- The - The fundamental counting principlefundamental counting principle can be used to count the number of can be used to count the number of possible outcomes given an event that possible outcomes given an event that can happen in some number of ways can happen in some number of ways followed by another event that can followed by another event that can happen in some number of different happen in some number of different ways.ways.


SummarySummary:: So when should I use a So when should I use a tree diagram or the fundamental tree diagram or the fundamental counting principle?counting 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 can can be used to:be used to:

(1) count the total number of (1) count the total number of outcomes.outcomes.

HomeworkHomework::--


Counting Outcomes

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