T—05/26/09—HW #71: Pg 713: 52 - 58; Pg 719: 12, 13, 24 - 29; Pg 734: 12 - 17; Pg 742: 10 – 13

52) 21700 54) 3797 56) perm, 210 58) comb, 924

12) .5 24) e: 13/60, t: 1/6 26) e: 59/120 t: ½

28) e: 29/40, t: 2/3

12) .047 14) .059 16) .020

10) .0000191 12) .0148

W—05/27/09—HW #72: Pg 706: 47, 61; Pg 713: 47, 51; Pg 734: 4 - 9; Pg 743: 19, 21; Pg 759: 28, 29

734:

4) .27 6) .2 8) .5

759:

28) 16

Chapter 12 Review

Given a hamburger, cheeseburger, and double-double for burgers,

fries light, fries well done, regular fries, animal fries for fries,

and a drink, shake, coffee, or milk for a beverage, how many different combos could you make?

Fundamental Counting Principle

If one event occurs in m ways, and another in n ways, the number of ways both can occur is m • n ways.

This can be extended to more than two events.

Burgers

(3 ways)

Fries

(4 ways)

Drinks

(4 ways)X X = 48 ways

Repetition versus non-repetition

If you had to make a 5-digit ID number where the first number must be 0 or 1, how many possible ID numbers are there:

A) If digits can repeat.

B) If digits cannot repeat.

How many options are there for the first blank?

2

How many options are there for the 2nd blank?

10

Don’t forget 0

10 10 10 =20,000

How many options are there for the first blank?

2

How many options are there for the 2nd blank?

9

Why? Because you can’t repeat a number you used before, so there is one less option.

8 7 6 =6,048

n! is read as “n factorial” and is represented by: (obligatory shout joke)

n(n – 1)(n – 2)…3•2•1 = n!

Or basically, 5! = 5•4•3•2•1 = 120

!7

5040

1234567

!6

!10

123456

12345678910

5040

Don’t multiply it all out, use some common sense.

For factorials on your calculator, it’s “Math” “PRB” “!”

0! = 1 IMPORTANT

There are 10 people taking a test, with no one getting the same score.

How many different ways can the students be ranked?

How many different ways can the students finish in first or second?

10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 3,628,800

10 • 9 = 90

This second example is special. There is a formula for an example like this:

Permutations of n objects taken r at a time

The number of permutations of r objects taken from a group of n distinct objects is denoted by nPr and is given by:

)!(

!

rn

nPrn

What this is saying is if I have n things to pick from, and pick r items, how many different ways can I arrange them?

90910!8

!10

)!210(

!10210

P

There are 7 food places Mr. Kim wants to visit, but he can only visit 4 of them. In how many different orders can Mr. Kim visit all these places?

8404567!3

!7

)!47(

!747

P

Calculator: press 7 first, then “math” “PRB” “nPr”, then 4.

There are 16 players on varsity baseball, and 9 spots on a line-up, how many different ways can the coach arrange to have all the batters hit?

200,347,151,4

8910111213141516

!7

!16

)!916(

!16916

P

We learned that permutations is when order is important. When order is NOT important, the number of different possibilities are called COMBINATIONS.

Permutation: How many different 9 man batting orders can you make with 16 players?

Combinations: How many different combinations of 9 players can play if you have 16 players? (Batting order doesn’t matter)

Permutation: Given 10 classes to choose from, how many 5 period schedules could you make?

Combinations: Given 10 classes, how many combinations of 5 classes could you make? (The order of classes don’t matter)

)!(!

!

rnr

nCrn

Combinations of n objects taken r at a time

The number of combinations of r objects taken from a group of n distinct objects is denoted by nCr and is given by:

35C 1025!2!3

!5

)!35(!3

!5

n how many to choose from

r how many you pick

13 topping options, how many pizzas can be made with AT MOST 2 toppings.

!0)!013(

!13

.2

:92

toppingsmostatwith

nscombinatioPizza

)2()1()( toppingstoppingToppingsNo 013C 213C

!2)!213(

!13

1 13

113C

!1)!113(

!13

78

When all outcomes are equally likely, the THEORETICAL PROBABILITY that an event A will

occur is:

outcomesofnumbertotal

AinoutcomesofnumberAP )(

Things to know:

Prime numbers are numbers where the factors are 1 and itself

Perfect square are numbers with “nice” square roots, like 25

Factors are numbers that divided evenly with a number.

Multiples are if you multiply a number, like multiples of 3 are 3, 6, 9, 12, etc

Sum means add

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Sum dice chart

An odd sum

A prime number The number five

choices

sumoddoddP

36

18)(

2

1

choices

primesprimeP

36

15)(

12

5

choices

fivesumfivesumP

36

4)(

9

1

)()()()(

:

,

BandAPBPAPBorAP

isBorAofyprobabilitthethen

eventstwoareBandAIf

)()()(

:

,

BPAPBorAP

isBorAofyprobabilitthe

thenexclusivemutuallyareBandAIf

Mutually exclusive – Nothing in common. P(A and B) = 0

)7()4()74( PPorP

52

4

52

4

52

8

Compound – Something in common

)(

)()()(

HeartandJackP

HeartPJackPHeartorJackP

52

1

52

13

52

4

13

4

52

16

)()()()(

:

,

BandAPBPAPBorAP

isBorAofyprobabilitthethen

eventstwoareBandAIf

)()()(

:

,

BPAPBorAP

isBorAofyprobabilitthe

thenexclusivemutuallyareBandAIf

)(

)(

)(

)(

BandAP

BorAP

BP

AP

Fill in the blanks, say if it is mutually exclusive or not

)(

)(

)(

)(

BandAP

BorAP

BP

AP

Probability of drawing a heart or a face card?

Probability of drawing a spade or club?

Probability of drawing an even number?

)( faceorHeartP )()()( faceandHeartPFacePHeartP

52

13

52

12

52

3

26

11

52

22

)()( bCluPSpadeP

52

13

52

13

2

1

52

26

)( bCluorSpadeP

)(evenP )10()8()6()4()2( PPPPP

52

4

52

4

52

4

13

5

52

20

52

4

52

4

Independent One event has NO effect on the other.

Examples: Flipping a coin.

Spinning a wheel.

Picking a card then putting it back.

Dependent One event HAS AN effect on the other.

Examples: Picking a card and NOT putting it back.

Picking a marble out of a bag and not putting it back.

Check hw

If A and B are INDEPENDENT events, then the probability that both A and B occur is:

P(A and B) = P(A) * P(B)

If A and B are DEPENDENT events, then the probability that both A and B occur is:

P(A and B) = P(A) * P(B|A)

That line means probability of B considering A already happened. This is called the CONDITIONAL PROBABILITY OF B given A.

Check hw

Classic Case involves Cards.

Terminology:

With replacement You pick the card, you put it back, so the previous choice DOES NOT affect the next one.

Without replacement You pick the card, you DO NOT put it back, so the previous choice DOES affect the next one.

W\ Replacement W\O Replacement

First card Ace

Second card Face

)()( FacePAceP

52

4

52

12

169

3

2704

48

)|()( AceFacePAceP

52

4

51

12

221

4

2652

48

First card Heart

Second card Heart

)()( HeartPHeartP

52

13

52

13

16

1

2704

169

)|()( HeartHeartPHeartP

52

13

51

12

17

1

2652

156

Perfect vs Imperfect information

Application to Hold ‘Em, I do not endorse it

W\ Replacement

?)(

5.)(

2.)(

BandAP

BP

AP

)(5.2. BandAP

?)(

1.)|(

4.)(

BandAP

ABP

AP

)(1.4. BandAP

W\O Replacement

)(1. BandAP

?034.)(

?)(

2.)(

BandAP

BP

AP

034.)(2. BP

17.)( BP

)(04. BandAP

1.)(

2.)|(

?)(

BandAP

ABP

AP

1.2.)( AP

)(5. APCheck hw

p. is each trialon success ofy probabilit thewhere

p)-(1pC successes)P(k

is successesk exactly ofy probabilit

then trials, of consisting experiment binomial aFor

yProbabilit Binomial Finding

k-nkkn

Coin toss

Multiple Choice (SAT style)

Inequalities with k, n = #, p = #

• T—05/26/09—HW #71: Pg 713: 52 - 58; Pg 719: 12, 13, 24 - 29; Pg 734: 12 - 17; Pg 742: 10 - 13

• W—05/27/09—HW #72: Pg 706: 47, 61; Pg 713: 47, 51; Pg 734: 4 - 9; Pg 743: 19, 21; Pg 759: 28, 29