Section 5.1.A

Basic Concepts of Probability

Today, you will learn to…

> identify the sample space of a probability experiment

Sample Space

all possible outcomes of a probability experiment

Roll a die: { 1, 2, 3, 4, 5, 6 }Flip one coin: { T, H }

Gender of one child: { M, F }

Event one possible outcome of a

trial in an experiment

Roll an even number { 2, 4, 6 }

Get tails when flipping a coin { H }

Draw a queen from a deck of cards{Q, Q, Q, Q}

If P(E) = 0 the event is impossible

the event is certainIf P(E) = 1

0 < P(E) < 1 or

0% < P(E) < 100%

Can probability be…

zero?

- 2%?

2.7 ?

¾ ?

YES

NO, negative

NO, greater than 1

YES

A probability experiment consists of tossing a coin and then rolling a six-sided die. Identify the sample

space.

H T

1 2 3 4 5 6 1 2 3 4 5 6

The sample space has 12 outcomes.

{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Draw a tree diagram showing the sample space of the gender sequence of a family with 3

children.B G

B G B G

B G B G B G B G

{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}

Draw a tree diagram showing the sample space of 4 days of weather (sunny or rainy).

S R

S R S R

S R S R S R S R

S R S R S R S R S R S R S RS R

S R

S R S R

S R S R S R S R

S R S R S R S R S R S R S RS R

List the sample space of 4 days of weather (sunny or rainy).

SSRS, SSSS, SSSR, SSRR,

S R

S R S R

S R S R S R S R

S R S R S R S R S R S R S RS R

{SSSS, SSSR, SSRS, SSRR, SRSS, SRSR, SRRS, SRRR,RSSS,RSSR, RSRS, RSRR,RRSS, RRSR, RRRS, RRRR}

16 different possibilities

Simple Event consists of one single

outcome

Simple: 65 in tall

Not Simple: over 6’ tall

Decide whether the event is simple or not.

the student's age is between 18 and 23

the student’s age is 20

not simple

simple

Decide whether the event is simple or not.

the student scored an 85%

the student scored a B

not simple

simple

Let’s Practice!!!!

Lesson 5.1.BTypes of Probability

Theoretical probability

Subjective probabilityExperimental probability

Theoretical probabilityis used when we already have the data we need to

find the probability.

dice, raffle, cards, coins, etc.

P (selecting a 7 of diamonds) =

P ( selecting a diamond) =

You select a card from a standard deck. Find the probability of the

following.

152

13 52

=

=

0.019 =

2%

25%

0.25 =

12 16

6 16

P(R or G) =

P (not R) =

A bag contains 16 marbles: 10 blue (B), 4 red (R), and 2 green (G). One marble

is randomly drawn from the bag.

0.75

=

=

=0.375

=

38%

75%

Experimental Probabilityis based on data collected

in an observation or experiment .

You actually do an experiment to find the probability

An insurance company analyst determines that in every 200 claims,

4 are fraudulent. What is the probability that the next claim the company processes is fraudulent?

4 200 = 0.02 = 2%P(fraudulent)=

A pond contains 3 types of fish.You catch 40 fish and record the

type. You catch 13 bluegill, 17 redgill, and 10 catfish. If you

throw all of the fish back and catch another fish, what is the probability that it is a catfish?

10 40

= 0.25 = 25%P(catfish)=

Toss a coin 10 times and count the number of times you get heads.

Find the class experimental probability.

Coin Tossing Probability Experiment

P(getting heads) = 105

= 50%

theoretical probability

If you repeat a probability experiment over and over, the

experimetnal probability of an event will equal the theoretical probability

of the event.

Law of Large Numbers

P(15 - 24 years old) =

Employee data was collected Employee Age Frequency, f

15 – 24 54

25 - 34 366

35 - 44 233

45 - 54 180

55 - 64 125

65 and over 42

541000

=0.054

=5%

Subjective probability

Results from intuition, educated guesses, and

estimates.

Subjective probabilityA doctor may feel that a patient has

a 90% chance of a full recovery.

A business analyst may predict that there is a 0.25 chance of

decreased sales.

A weather reporter makes an educated guess that there is a

20% chance of rain today.

Theoretical, Experimental, or Subjective?

The probability of your phone ringing during class is 0.15

subjective probability because it is most likely based on an

educated guess

Calculate the probability of the event & its complement.

E: Pick a red card

E: Roll a 3 or greater

P(red) =

P(black) =

50%

P(>3) = =67% P(<3) = 33%

50%E’:Pick a black

E’: Roll a 1 or 2

46

The probability that a voter chosen at random will vote

republican is 45%

statistical probability because it is most likely based on a survey of a

sample of voters

Theoretical, Experimental, or Subjective?

theoretical probability because you know the number of outcomes and

each is equally likely

The probability of winning a 1000-ticket raffle with one

ticket is 1 in 1000.

Theoretical, Experimental, or Subjective?

The complement of event E (E’) is the set of all outcomes in a

sample space that are

NOT included in event E.

P(E) + P(E’) = 1

P(E) + P(E’) = 100%

Identify the complement of the event. Give both probabilities.

E: Roll a 4

E: Roll an odd number

E’:

E’:

P(E) = P(E’) =

P(E) = P(E’) =

16

56

3612

12

roll a 1,2,3,5,or 6

roll an even

Practice Time!!!!

Lesson 5.2.A

Fundamental Counting

PrincipleToday, we will learn to… > use the Fundamental Counting

Principle

How many ways I can put together outfits with 3 pairs of

pants (jeans, black, and tan) and 4 shirts (white, purple, red,

and teal)?

Jeans

Black

Tan

12 outfits

whitepurpleredteal

whitepurpleredteal

whitepurpleredteal

Find the number of ways I can put together a sundae with 3 kinds of ice cream (van, choc, swirl), 3 different

toppings (caramel, strawberry, butterscotch) and 3 different crunchy toppings (choc chips, M&Ms, oreo).

Van Choc Swirlcaram

el

strawb

erry

bu

tterscotch

caramel

strawb

erry

bu

tterscotch

caramel

strawb

erry

bu

tterscotch

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

Choc. C

hipM

&M

Oreo

27 different sundaes

The Fundamental Counting Principle

If one event can occur in A ways, another event can occur in B ways, another event can occur in C ways,and so on, then the total number of

possible outcomes isA∙B∙C∙…

The rule can be extended for any number of events.

You are purchasing a new car. You can choose from 4 different manufacturers,

3 different car sizes, and 6 different colors. How many different ways can you select one manufacturer, one car

size, and one color?

72 cars_·_·_ =4 3 6

The access code for a car’s security system consist of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be

used only once and not repeated?

__·__·__·__ = 504010 9 8 7

The access code for a car’s security system consist of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be

repeated?

__·__·__·__ =10 10 1010 10,000

How many license plates can you make if a license plate consists of six

letters that cannot be repeated?

__·__·__·__·__·__=26 165,765,60025 24 23 22 21

How many license plates can you make if a license plate consists of six letters that can be repeated?

__·__·__·__·__·__=26 308,915,77626 26 26 26 26

Let’s Practice!!!!

Lesson 5.2.B

Permutations

Today, we will learn to… > count the number of possible outcomes using a permutation

A permutation is an ordered arrangement of objects. The number of different

permutations of n objects is n!

5! = 5·4·3·2·1

3! = 3·2·1

2! = 2·1

The starting lineup for a baseball team consists of nine players.

How many different batting orders are possible using the starting

line up?

9! =

9·8·7·6·5·4·3·2·1 =

362,880

The teams in the National League Central Division are listed. How many different final standings are possible?

Chicago Cubs Cincinnati RedsHouston Astros Milwaukee BrewersPittsburgh Pirates St. Louis Cardinals

6! =6·5·4·3·2·1 = 720

The starting lineup for a baseball team consists of nine players. How many different batting orders are possible

from a group of 20 players?

__ · __ · __ · __ · __ · __ · __ · __ · __20 19 18 17 16 15 14 13 12

If you use 20!, how could you reduce out the 11 through 1?= 60,949,324,800

20! 11!

The number of permutations (ordered arrangements) of n different objects taken

r at a time is

nPr = n!

(n – r)!

9P6 = 9!

3!

Find 43P3 = 74,046

Find 26P10

= 19,275,223,970,000

43!40!

26!16!

43 · 42 · 41

26 · 25 · 24 · 23 · 22 · 21 · 20 · 19 · 18 · 17 =

Let’s Practice!!!!

Lesson 5.2.C Distinguishable

PermutationsIn this lesson, we will learn to…> count the number of possible outcomes using distinguishable permutations

> count the number of possible outcomes using combinations

B B G B G B B

What are Distinguishable Permutations?

Look at these arrangements…

B B B B BG G

Do you notice a difference?

How many ways can we order a group of 7?

If we have 4 A’s, 2 B’s, and 1 C, there are 105 distinguishable

permutations.

5040 ordered arrangements

7!

= 105

7! = 7·6·5·4·3·2·1

4 2

2!)(4!

The number of distinguishable permutations of n objects with A of one type,

B of another type, and so on is…

n!

A! · B! · C! · D! ···

A developer is planning a new subdivision that will consist of 6 one-story houses, 4 two-story

houses, and 2 split-level houses. In how many distinguishable ways

can the houses be arranged?

__! _! · _! · _!

= 13,86012

6 4 2

A contractor wants to plant 6 oak trees and 9 maple trees along

the street. If the trees are spaced evenly apart, in how many

distinguishable ways can they be planted?

__! _! · _!

= 5005156 9

What are Combinations?

Combinations tell us how many ways a GROUP of different things

can be formed.

Order does NOT matter.

With the letters A, B, C, and D there are 12 different ways to

select two letters.

PermutationsAB BA AC CA AD DABC CB BD DB CD DC

CombinationsBA AC CA AD DA

BC CB BD DB CD DCAB

You want to buy 3 CDs from a selection of 5 CDs. Label the CDs

A, B, C, D, and E. Write the 10 ways to make your selection.

ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE

Order doesn’t matter so ABC is the same as BAC.

Choose 3 CDs from a choice of 5 CDs

2 · 1

5 · 4 · 3 · 2 · 1

3!

because order doesn’t matter

combinations are like distinguishable permutations

=5 !

3 ! · 2 !

The number of combinations of r objects selected from a group

of n objects is…

n!

(n – r)! · r! nCr =

20!

(16! · 4!)20C4 =

The department of transportation receives 16 bids for a project. They plan to hire 4 companies.

How many different ways can 4 companies be selected from the

16 bidding companies?

__!__! · __!

= 182016

12 416C4 =

The manager of an accounting department wants to form a

3-person committee from the 16 employees in the department.

In how many ways can the manager do this?

__!__! · __!

= 56016

13 316C3 =

How many ways can a jury of 5 men and 4 women be selected from

12 men and 14 women?

(12C5)(14C4) =

792,792 ways

__!__! · __!

127 5

__!__! · __!

1410 4

Let’s Practice!!!!

A bag of M&M’s contains lilac, pink, and yellow candy. Construct a tree

diagram of the possible outcomes of picking 3 M&M’s out of the bag.

L P Y

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

total possibilities

L P Y

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

Find the probability of getting 2 yellow.

Sets with 2 yellow? 6

27= 0.222 = 22%

* ** ***

total possibilities

L P Y

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

Find the probability of getting at least 1 lilac.

Sets with at least 1 lilac? 19

27= 0.704 = 70%

*** ****** **** * *** * *

total possibilities

L P Y

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

L P Y

LPY LPY LPY

Find the probability of getting 1 lilac, 1 yellow, and 1 pink.

Sets with one of each? 6

27= 0.222 = 22%

* * * * * *

In how many distinguishable ways can the letters in MISSISSIPPI be arranged?

__! __! · __! · __!

= 34,650114 4 2

If an M, 4 S’s, 4 I’s, and 2 P’s are randomly arranged, what is the probability that they will spell Missippippi?

1 34,650

=0.00002886= 0%

A test consists of 20 True-False questions. How many different ways are there to answer this test?

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 20 = 1,048,576How many of these ways are correct? 1

What is the probability of guessing and getting a 100 on the test?

11,048,576

= 0.000000954 = 0%

· · · · · · · · · · · · · · · · · · ·

Let’s Practice!!!!