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Probability. Unit Essential Question: How does probability relate to the field of statistics and what implications does this have?. Probability. P1. Counting and Probability. How is probability defined?. Activation.

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slide1

Probability

Unit Essential Question: How does probability relate to the field of statistics and what implications does this have?

p1 counting and probability

Probability

P1. Counting and Probability

How is probability defined?

slide3

Activation

A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad?

How can you be sure you have all the combinations

slide5

Tree Diagrams

Steps: Vertically list the choices for the first

event.

Draw branches for each choice for

the second event coming from each choice of the first.

Repeat for each event.

slide6

Tree Diagrams Example

A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad?

ranch

mushrooms

Italian

ranch

beans

Italian

Lettuce

ranch

cheese

Italian

ranch

mushrooms

Italian

Spinach

ranch

beans

Italian

cheese

ranch

Italian

slide7

Fundamental Counting Principle

Tree diagrams are helpful, but what if there are many events and many choices?

The fundamental counting principle is a mathematical method to figure out the number of ways a compound event may occur.

# of outcomes for event 1 *# of outcomes for event 2*…

slide8

Fundamental Counting Principle Ex

Mr. Giuliano has 2 pairs of shoes, 3 pairs of pants, 10 shirts, and 4 ties. How many different outfits can he make before he repeats?

slide9

Permutations

Permutations: all the possible ways a group of objects can be arranged or ordered.

Order DOES matter!

Example:

There are four different books to be placed in order on a shelf. A history book (H), a math book (M), a science book (S), and an English book (E). How many ways can they be arranged?

H, M, S, E

M, E, S, H

S, M, E, H

E, M, S, H

H, M, E, S

M, E, H, S

S, M, H, E

E, M, H, S

H, S, E, M

M, S, H, E

S, H, M, E

E, H, M, S

H, S, M, E

M, S, E, H

S, H, E, M

E, H, S, M

H, E, M, S

M, H, E, S

S, E, M, H

E, S, M, H

H, E, S, M

M, H, S, E

S, E, H, M

E, S, H, M

slide10

Factorials

Sometimes we do not need to use all the objects in a given set

EX: Arrange 3 of the 4 books

EX: Arrange 2 of the 5 books

slide11

Factorials

The product of all the numbers from thegiven number down to 1

n! = n (n-1) (n-2) (n-3) ••• 3 (2) (1)

Definition 1! = 1 and 0! = 1

slide12

Permutation Formula

A permutation of n objects r at a time follows the formula

Where:

n=total objects in the set

r=the objects to be used

So if we are to

arrange 3 of 7 objects….

slide14

P2 Repetitions and Circular Permutations

Essential Question: What is the difference between replacement and repetition?

slide15

Activation

What does replacement mean—how might that relate to probability?

slide16

Replacement

Replacement—using the same object again (nr)

Example:

The keypad on a safe has the digits 1- 6 on it how many:

a) four digit codes can be formed

_____ _____ _____ _____

b) four digit codes can be formed if

no 2 digits can be the same

_____ _____ _____ _____

slide17

Repetitions

Repetition—occurs when you have identicalitems in a group

Example:

Find all arrangements for the letters in the word

TOOL

TOOL OLOT LOTO

TOLO OLTO LOOT

TLOO OTOL LTOO

OTLO

OOTL

OOLT

Since the o’s are identical, we eliminate all duplicate possibilities. Hence, there are only 12 arrangements.

slide18

Formula for Repetitions

where s and t represent the

number of times an item is

repeated

EXAMPLE:

How many ways can you arrange the letters in BANANAS

slide19

Circular Permutations

Circular Permutation—arranging items in a circle when no reference is made to a fixed point

ExCircular Permutation—arranging items in a circle when no reference is made to a fixed point

Example:

How many ways can you arrange the numbers 4 guests around a table?

ample:

How many ways can you arrange the numbers 1-4 on a spinner?

We would expect 4! Or 24 ways but we only have 6

Circular pCircular permutations are always (n-1)!

ermutations are always (n-1)!

2

1

1

2

4

4

3

3

1

1

1

1

1

2

2

1

E

C

A

F

G?

B

?

D

3

4

3

4

3

3

2

2

4

2

2

4

4

4

3

3

D

slide20

Homework

Worksheet #2

slide21

P3. Combinations

Essential Question: How can you tell the difference between a permutation and a combination?

slide22

Activation

When a recipe says combine the following ingredients—what does that mean?

How is that different than add the following ingredients one at a time?

slide23

Combinations

Combinations: the number of groups that can beselected from a set of objects

The order in which the items in the group are selected does not matter!

Notation: The number of combinations of a set of n objects taken r at a time is

n=total objects r=number of objects being used

slide24

Combinations

Example: How many three person committees can be formed from a group of 4 people—Joe, Jim, Jane, and Jill

Formula:

Joe, Jim , Jill

Joe, Jill, Jane

Joe, Jim Jane

Is Joe, Jane, Jim

A different committee?

Jim, Jane, Jill

slide25

Homework

Worksheet Number 3

slide26

P4. Probability

Essential Question: How is an independent event defined?

slide27

Activation

How are these two situations different:

There are 5 marbles in a bag what is the probability of getting a red one on the second try if three are red and 2 are blue if you choose a marble and replace it?

There are 5 marbles in a bag what is the probability of getting a red one on the second try if three are red and 2 are blue if you choose a marble but keep it out?

slide28

P4. Probability

Independence: When the outcome of one event does not impact the outcome of another event.

slide29

Probability

If all outcomes are successful, the probability will be 1

If no outcomes are successful, the probability will be 0

So…Probability is 0 ≤ P ≤ 1

slide30

Examples

What is the probability of getting an ace from a deck of 52 cards?

What is the probability of rolling a 3 on a 6 sided die?

slide31

More Examples

What is the probability of rolling an even number?

slide32

More Examples

  • What is the probability of getting a total of 5 when a red die and green die is rolled?
slide33

More Examples

  • What is the probability of getting 2 spades when 2 cards are dealt at the same time?
slide34

Home Work

  • Worksheet 4
slide35

P5. Compound Probability

  • Essential Question: What is meant by compound probability?
slide36

Activation

  • What is a compound sentence in English?
  • What about in Algebra?
  • How might this impact probability?
slide37

Compound Probability with “Or”

  • If events are exclusive, they have no overlap.
  • Ex. What is the probability of rolling a 3 or a 5?
  • “Or” implies addition.
  • P(A or B)=P(A)+P(B)
slide38

Compound Probability with “Or”

  • What is the probability of drawing an ace or a heart?
  • # of cards_____
  • # of hearts____
  • # of aces_____

# of hearts that are aces____

  • When events are inclusive, they have overlap that must be considered.
  • OR: P(A or B) = P(A) + P(B) – P(A and B)
slide39

Compound Probability with “And”

  • “And” implies multiplication.
  • P(A and B)=P(A)*P(B)
  • Ex. What is the probability of rolling a 4 and drawing an ace?
  • Are these events dependent or independent?
slide40

Homework

  • Worksheet 5
slide41

P6. Binomial Expansion

Essential Question: What is binomial expansion and how does it relate to probability?

slide42

Activation

Look at it in terms a algebra first:

Expand (a + b)3

=(a + b)(a + b)(a + b)

=(a2 + 2ab + b2)(a + b)

=a3 + 2a2b + ab2 + a2b + 2ab2 + b3

=a3 + 3a2b + 3ab2 + b3

What if this had been raised to the 10th power?

slide43

Binomial Expansion

(a + b)01

(a + b)1 1a + 1b

(a + b)2 1a2 + 2ab + 1b2

(a + b)31a3 + 3a2b + 3ab2 + 1b3

The pattern for the variables is simple start with the highest power on the 1st variable and count down, start with 0 on the second variable and count up

The pattern on the coefficients is less obvious but it follows Pascal’s triangle

  • 1
  • 1
  • 1 2 1
  • 3 3 1
  • 4 6 4 1
slide44

Binomial Expansion

n = the exponent

r = the position of the term – 1 (for the2nd term r = 1)

a = the part in the 1st half of the ()

b = the part in the 2nd half of the () including the sign

Find the 7th term of (4x –y2)9

n = 9

r = 7-1

slide47

How does this relate to probability?

Let a = the probability that an event did not occur

b = the probability that it did occur

n = the # of trials

r = # of success

Works for any problem which is a dichotomy—something that either happens or does not happen

Since total probability = 1

Then p’ = 1-p

slide48

Example of how this relates to probability

What is the probability that a family with 9 children has 7 girls?

a = .5

b = .5

n = 9

r = 7

slide49

Example of how this relates to probability

What is the probability that a family with 6 children has at least 4 boys?

a = .5

b = .5

n = 6

r = 4 or 5 or 6

slide50

Example of how this relates to probability

What is the probability that a 300 hitter will hit at least 4 times in 5 hits

(a 300 hitter hits 300/1000 times)