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

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

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.

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

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*…

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?

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

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

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

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….

P2 Repetitions and Circular Permutations

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

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

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

_____ _____ _____ _____

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.

where s and t represent the

number of times an item is

repeated

EXAMPLE:

How many ways can you arrange the letters in BANANAS

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

Worksheet #2

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

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?

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

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

Worksheet Number 3

Essential Question: How is an independent event defined?

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?

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

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

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?

What is the probability of rolling an even number?

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

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

- Worksheet 4

- Essential Question: What is meant by compound probability?

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

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)

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)

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?

- Worksheet 5

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

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?

(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

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

Examples of Binomial Expansion

Find (2x + 3)4

Examples of Binomial Expansion

Find (a - b)9

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

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

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

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)

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