The Counting Principle Counting Outcomes Have you ever seen or heard the Subway or Starbucks advertising campaigns where they talk about the 10,000 different

The The Counting Counting PrinciplePrinciple

Counting OutcomesCounting 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 COUNTING PRINCIPLE(2) THE COUNTING PRINCIPLE

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

http://www.youtube.com/watch?v=8WdSJhEIrQk&safe=active


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)

The Tree DiagramThe Tree Diagram

A Different WayA Different Way

Example 1Example 1:: The Counting Principle.The 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)


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

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.

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

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


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?

You use a tree diagram because you You use a tree diagram because you want a specific outcome … not the want a specific outcome … not the TOTAL number of outcomes.TOTAL number of outcomes.


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 3Example 3:: The Counting Principle.The Counting Principle.

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 total digits from 0 to 9, what is the total number of outcomes?number of outcomes?

You use the Counting Principle because You use the Counting Principle because you want the total number of you want the total number of outcomes. How many possible digits outcomes. How many possible digits are from 0 to 9?are from 0 to 9?


Example 3Example 3:: The Fundamental The Fundamental Counting Principle.Counting Principle.

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 total number of from 0 to 9, what is the total number of possible outcomes?possible outcomes?

# 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

What chance would you have to win if you played one What chance would you have to win if you played one time?time?

Guided PracticeGuided Practice:: Tree or Counting Tree or Counting Principle?Principle?

(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?

Example 1Example 1

You are purchasing a new car. Using You are purchasing a new car. Using the following manufacturers, car sizes the following manufacturers, car sizes and colors, how many different ways and colors, how many different ways can you select one manufacturer, one can you select one manufacturer, one car size and one color?car size and one color?

Manufacturer: Ford, GM, ChryslerManufacturer: Ford, GM, ChryslerCar size: small, mediumCar size: small, mediumColor: white(W), red(R), black(B), Color: white(W), red(R), black(B),

green(G)green(G)

SolutionSolution

There are three choices of There are three choices of manufacturer, two choices of car manufacturer, two choices of car sizes, and four colors. So, the sizes, and four colors. So, the number of ways to select one number of ways to select one manufacturer, one car size and one manufacturer, one car size and one color is:color is:

3 ●2●4 = 24 ways. 3 ●2●4 = 24 ways.

Ex. 2 Using the Fundamental Ex. 2 Using the Fundamental Counting PrincipleCounting Principle The access code for a car’s security The access code for a car’s security

system consists of four digits. Each system consists of four digits. Each digit can be 0 through 9. How many digit can be 0 through 9. How many access codes are possible if each access codes are possible if each digit can be repeated?digit can be repeated?

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The Counting Principle Counting Outcomes Have you ever seen or heard the Subway or Starbucks advertising campaigns where they talk about the 10,000 different