Chapter 15 Counting and Probability Section 15-1 Counting Problems and Permutations

Chapter 15

Counting and Probability

Section 15-1

Counting Problems and Permutations

Definition Outcome

The result of a succession of events

Definition Event

A subset of all possible outcomes.

A compound event is composed of two or more events

In situations where we consider the combinations of items, or the succession of events such as flips of a coin or the drawing of cards, each result is called an outcome.

An event is a subset of all possible outcomes. When an event is composed of two or more outcomes, such as choosing a card followed by choosing another card, we have a compound event

Theorem 15-1

The fundamental counting principle

In a compound event in which the first event can happen in n1 ways and the second event in n2 different ways and so on, and the kth event can happen in nk different ways, the total number of ways the composed event can happen is

1 2 ... kn n n

Total possible number of dinners = 2(4)(2) = 16

Definition Permutation

A permutation of a set of n objects is an ordered arrangement of the objects.

Theorem 15-2

The total number of permutations of a set of n objects is given by

1 2 3 2 1( ) ( ) ... !n nP n n n n

The set of n objects taken n at a time

Find the number of possible arrangements of the set {a,b,c,d}

Find the number of possible arrangements of the set {3,4,7}

Possible Codes:

_____ _____ _____ _____

_____ _____ _____

_____ _____

_____

2 2 2 2

2 2 2

2 2

2

42

32

22

12

30

Eight students are to be seated in a classroom with 11 desks.

Calculate the number of seatings by choosing one of the desks for each student.

Calculate the number of seatings by choosing one student for each of the desks, after increasing the number of students to 11 by imagining that there are 3 “invisible” students (who are, of course, indistinguishable).

HW #15.1Pg 648-649 1-51 Odd

Chapter 15Counting and Probability

15-2

Permutations For Special Counts

Objective: Find the number of permutations of n objects taken r at a time with replacement.



Objective: Find permutations of a set of objects that are not all different.

How many different words (real or imaginary) can be formed using all the letters in the word FREE?

Objective: Find permutations of a set of objects that are not all different.

How many different words (real or imaginary) can be formed using all the letters in the word TOMORROW?

Eight students are to be seated in a classroom with 11 desks.

Calculate the number of seatings by choosing one of the desks for each student.

Calculate the number of seatings by choosing one student for each of the desks, after increasing the number of students to 11 by imagining that there are 3 “invisible” students (who are, of course, indistinguishable).

Three Types of Permutations

How many ordered arrangements are there of 5 objects a, b, c, d, e choosing 3 at a time without repetition?

How many ordered arrangements are there of 5 objects a, b, c, d, e choosing 3 at a time with repetition?

How many ordered arrangements are there of 5 objects a, a, d, d, e choosing 5 at a time without repetition?

If you have a group of four people that each have a different birthday, how many possible ways could this occur?

Objective: Find circular permutations

Objective: Find circular permutations

Find the number of possibilities for each situation.

A basketball huddle of 5 players

Four different dishes on a revolving tray in the middle of a table at a Chinese restaurant

six quarters with designs from six different states arranged in a circle on top of your desk

HW #15.2Pg 654-655 1-9, 11-19 Odd, 20-34

15-3

Combinations

Definition Combination

A Combination of a set of n objects is an arrangement, without regard to order, of r objects selected from n distinct objects without repetition, where r n.

Find the value of each expression.

4

2)a

5

2)b

)n

cn

0

)n

d

40

4)e

List all the combinations of the 4 colors, red, green, yellow and blue taken 3 at a time.

How many different committees of 4 people can be formed from a pool of 8 people?

How many ways can a committee consisting of 3 boys and 2 girls be formed if there are 7 boys and 10 girls eligible to serve on the committee?

How many ways can a congressional committee be formed from a set of 5 senators and 7 representatives if a committee contains 3 senators and 4 representatives?

In the California Mega Lottery you choose 5 different numbers form 1 to 56 and then choose 1 number from 1 to 46 for a total of 6 numbers. How many ways can you choose these 6 numbers?

Winning the Lottery

A hamburger restaurant advertises "We Fix Hamburgers 256 Ways!“ This is accomplished using various combinations of catsup, onion, mustard, pickle, mayonnaise, relish, tomato, and lettuce. Of course, one can also have a plain hamburger. Use combination notation to show the number of possible hamburgers, Do not evaluate.

HW #15.3Pg 658 1-30

15-4 Binomial theorem

Lesson

• Pascal’s Triangle and Relation to Combinations

• Do some expansions

• Solve for x

• Straight from the book

• HW 15.4 Pg 662-663 1-28

HW 15.4Pg 662-663 1-28

15-5 Probability

Objective: Compute the probability of a simple event.

Definition Event

The result of an experiment is called an outcome or a simple event. An event is a set of outcomes, that is, a subset of the sample space.

Definition Sample Space

The set of all possible outcomes is called a sample space.


For example

The experiment: Throwing a dart at a three-colored dart board

Sample space: Three outcomes, {red, yellow, blue}.

An Event: Hitting yellow.

When the outcomes of an experiment all have the same probability of occurring, we say that they are equally likely.


12 352 13

1 3 1 61

6 6 2 6) ) )a b c


3 16 2

1 18 40320

)!

a

4

2 38 14

2

)b

1 14 24

)!

a

2

2 14 6

2

)b



The answers are all determined if you know which questions you will answer correctly and which you will answer incorrectly.



HW #15.5Pg 665-666 1-22

15-6 Probability of Compound Events

When you consider all the outcomes for either of two events A and B, you form the union of A and B. When you consider only the outcomes shared by both A and B, you form the intersection of A and B. The union or intersection of two events is called a Compound Event

Compound Events are considered Mutually Exclusive if the intersection of the two events is empty.

Mutually Exclusive EventsTwo Events are mutually exclusive if they cannot occur at the same time

Drawing a Face Card or an Ace are mutually exclusive

A card is randomly selected from a standard deck of 52 cards. What is the probability that it is an ace or a face card?

Example

Mutually Exclusive EventsTwo Events are mutually exclusive if they cannot occur at the same time

Drawing a Heart or a Face Card are not mutually exclusive

A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a heart or a face card?

Example

Objective: Find the probability that one event or another will occur.

If A and B are mutually

Exclusive, then P(AB) = 0

Examples mut/not 1

Event A: You draw a jack or a king on a single draw from a standard 52 card deck.

Event B: You draw a king or a diamond on a single draw from a standard 52 card deck.

( ) (king jack)P A ( ) (king diamond)P B

MutuallyExclusive

Not Mutually Exclusive

( ) ( ) ( )P A P k P j

4 4 2( )

52 52 13P A

( ) (k) ( ) ( )P B P P d P k d

4 13 1( )

52 52 52P B 4

13

A: Select an aceB: Select a face card.

A and B are mutually exclusive

SOLUTIONA: Select a heartB: Select a Face Card

A and B are NOT mutually exclusive

A standard six-sided number cube is rolled. Find the probability of the given event.

1. an even number or a one

2. a six or a number less than 3

3. an even number or number greater than 5

4. an odd number or number divisible by 3

3 1 4 26 6 6 3

1 2 3 16 6 6 2

3 1 1 3 16 6 6 6 2

3 2 1 4 26 6 6 6 3

Objective: Find the probability that one event and another event will occur.

Compute the probability of drawing a king and a queen from a well-shuffled deck of 52 cards if the first card is not replaced before the second card is drawn.

Independent/Dependent

Independent Events

The occurrence or no occurrence of one event does not effect the probability of the other.

P(A B) = P(A)P(B)

Dependent Events

The occurrence of the first event effects the probability of the other event.

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

By definition P(A | B) = the probability of A given that B has occurred.

Independent/Dependent

Independent Events

The occurrence or no occurrence of one event does not effect the probability of the other.

P(A B) = P(A)P(B)

Dependent Events

The occurrence of the first event effects the probability of the other event.

P(A B) = P(A) P(B | A) = P(B) P(A | B)

By definition P(A | B) = the probability of A given that B has occurred.

Examples Ind/Dep 1

Event A: You roll two dice. What is the probability that you get a 5 on each die?

Event B: What is the probability you draw 2 cards from a standard deck of 52 cards and get two aces?

( ) (5 5)P A ( ) (Ace Ace)P B

Independent Dependent

( ) (5) (5)P A P P 1 1 1

( )6 6 36

P A

nd st( ) ( ) (Aceon 2 |Aceon 1 )P B P Ace P

4 3 1( )

52 51 221P B

Examples Ind/Dep 1 -compare-sample-space

Event B: What is the probability you draw 2 cards from a standard deck of 52 cards and get two aces?

Sample Space Method:

#of ways todraw twoaces( )

#of ways todraw twocardsP B

4

2( )

52

2

P B

1221

( ) (Ace Ace)P B

Dependent

4 3 1( )

52 51 221P B

nd st( ) (Aceon 2 card|Aceon 1 card)P B

Examples Ind/Dep 2 -compare-sample-space

Compute the probability of drawing a king and a queen from a well-shuffled deck of 52 cards if the first card is not replaced before the second card is drawn.

Multiplication Rule Method:

P(king on 1st queen on 2nd) =

= P(king on 1st) P(queen on 2nd, given king on 1st)

4 452 51

4663

Sample Space Method:

P(king on 1st card queen on 2nd card) =

4,1 4,1

52,2

P P

P

4663

Multiplication rule order matters

Probability of a king and queen in that order

SOLUTIONA: Getting more than $500 on the first spinB: Going bankrupt on the second spin.

The two events are independent.

Find the probability of the given events.

1. An 8 on the first roll and doubles on the second roll?

2. An even sum on the first roll and a sum greater than 8 on the second roll.

3. Drawing a 5 on the first draw and a king on the second draw without replacement.

5 6 58

36 36 216( ) ( )P P Doubles

18 10 58

36 36 36 ( ) ( )P Even P Sum

4 4 16 45 5

52 51 2652 663( ) ( | )P P K

HW #15.6aPg 670-671 1-23 Odd

15-6 Day 2Probability of Compound Events

PROVE: If A and B are independent, then P(B | A) = P(B)

What is the probability that in a room of 40 people 2 or more people have the same birthday?

Rolling Dice Two six-sided dice are rolled. Find the probability of the given event.

The sum is even and a multiple of 3.

The sum is not 2 or not 12.


( 3) ( ) ( 3 | )P Even mult P even P mult even

18 6 6 1

36 18 36 6

Rolling Dice Two six-sided dice are rolled. Find the probability of the given event.

The sum is greater than 7 or odd.

The sum is prime and even.



A standard deck of cards contains 4 suits (heart, diamond, club, spade) and 13 cards per suit (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King). Suppose five cards are drawn from the deck without replacement. What is the probability that the cards will be the 10, jack, queen, king, and ace of the same suit?

4

1

5

5

52

5

0 28 0 41 0 16If . , . , and . , what is ?P A P B P A B P A B

HW #15.6bPg 670-671 2-22 Even, 24, 26-29

More Fun With Probability


Suppose 5 cards are drawn from a deck of 52 cards. What is the probability that you draw 5 Spades?

13

5 128752 2598960

5


Suppose 5 cards are drawn from a deck of 52 cards. What is the probability that you draw 2 cards with the same number and 3 other different cards?

13

1

4

2

12

3

4

1

4

1

4

1

52

5


Suppose 5 cards are drawn from a deck of 52 cards. What is the probability that you draw 3 cards with the same number and 2 other different cards?

13

1

4

3

12

2

4

1

4

1

52

5



HW#R-15Pg 685-686 1-22

Definition: Expected Value• In probability theory the expected value of a random

variable is the sum of the probability of each possible outcome of the experiment multiplied by its value.– Represents the average amount one "expects" to win per

game if bets with identical odds are repeated many times.

– Note that the value itself may not be expected in the general sense; It may be unlikely or even impossible.

– A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a “fair game”.

ExampleFind the expected value of X, where the values of X and their corresponding probabilities are given by the following table: xi 2 5 9 24

pi 0.4 0.2 0.3 0.1

SOLUTION

( ) 0.4 2 0.2 5 0.3 9 0.1 240.8 1.0 2.7 2.46.9

E X

Ten cards of a children’s game are numbered with all possible pairs of two different numbers from the set {1, 2, 3, 4, 5}. A child draws a card, and the random variable is the score of the card drawn. The score is 5 if the two numbers add to 5; otherwise, the score is the smaller number on the card. Find the expected score of a card.

Score Pairs5 (1, 4), (2, 3)1 (1, 2),(1, 3), (1, 5)2 (2, 4), (2, 5)3 (3, 4), (3, 5)4 (4, 5)

Two numbers can be selected from the five numbers in C(5, 2) = 10 ways

E(X) = 1(3/10) + 2(2/10) + 3(2/10) + 4(1/10) + 5(2/10) = 2.7

For example, an American roulette wheel has 38 equally possible outcomes. A bet placed on a single number pays 35-to-1 (this means that you are paid 35 times your bet and your bet is returned, so you get 36 times your bet). So the expected value of the profit resulting from a $1 bet on a single number is, considering all 38 possible outcomes:

Expected Value Roulette

which is about −$0.0526. Therefore one expects, on average, to lose over five cents for every dollar bet.

Expected Value Suppose you work for an insurance company, and you sell a $10,000 whole-life insurance policy at an annual premium of $290. Actuarial tables show that the probability of death during the next year for a person of your customer's age, sex, health, etc., is .001. What is the expected gain (amount of money made by the company) for a policy of this type?

Gain x Sample Point Probability

$290 Customer lives .999

$290-$10,000 Customer dies .001

If the customer lives, the company gains the $290 premium as profit. If the customer dies, the gain is negative because the company must pay $10,000, for a net "gain" of $(290 - 10,000).

Expected Value Gain x Sample Point Probability

$290 Customer lives .999

$290-$10,000 Customer dies .001

( )E xP x 290(.999) (290 10000)(.001)

280

The insurance company expects to make a $280 profit on the deal at the end of the first year.

Expected Value The chance of winning Florida’s pick six lottery is about 1 in 14,000,000. Suppose you buy a $1.00 lotto ticket in anticipation of the $7,000,000 jackpot. Calculate your expected net winnings.

Gain x Sample Point Probability

7,000,000 Win 1/14,000,000

-1 Lose 13999999/14000000

( )E xP x 1 13,999,9997,000,000 (0)

14,000,000 14,000,000

$0.50

The college hiking club is having a fundraiser to buy a new toboggan for winter outings. They are selling Chinese fortune cookies for 35 cents each. Each cookie contains a piece of paper with a different number written on it. A random drawing will determine which number is the winner of a dinner for two at a local Chinese restaurant. The dinner is valued at $40. Since the fortune cookies were donated to the club, we can ignore the cost of the cookies. The club sold 816 cookies. John bought 12 cookies.

1. What is the probability he will win

2. What is the probability he will loose?

3. What is the expected value of the game?

Binomial Probability ModelOn a TV quiz show each contestant has a try at the wheel of fortune. The wheel of fortune is a roulette wheel with 36 slots, one of which is gold. If the ball lands in the gold slot, the contestant wins $50,000. No other slot pays. What is the probability that the quiz show will have to pay the fortune to three contestants out of 100?

Binomial Probability• There are only two outcomes – Success or Failure

• There is a number of fixed trials

• All trials are independent and repeated under identical conditions

• For each trial the probability of success is the same and P(success) + P(Failure) = 1

• Looking for the P(r successes out of n trials)

(r succesesin n trials) ( ) ( ( ))r n rn

P P S P Fr

Wheel of Fortune Problem• Each of the 100 contestants has a trial so

there are 100 trails (n = 100)• The trials are independent (assuming a fair

wheel)• Only two outcomes win or lose• P(Success) = 1/36• P(Failure) = 35/36• P (Success) + P(Failure) = 1

3 100 3100 1 35(3succesesin100 trials)

3 36 36

P

A fair quarter is flipped three times. Find the following probabilities:

1. You get exactly three heads

2. You get exactly two heads

3. You get two or more heads

4. You get exactly three tails

3 03 1 1 13 2 2 8

2 13 1 1 32 2 2 8

2 1 3 03 31 1 1 1 1

2 2 2 3 2 2 2

0 33 1 1 10 2 2 8

Joe Blow has just been given a 10 question multiple choice quiz in history class. Each question has 5 answers of which one is correct. Since Joe has not attended class recently, he does not know any of the answers. Assuming Joe guesses on all 10 questions, find the indicated probabilities

1. Joe gets all 10 correct

2. Joe gets all 10 wrong

3. Joe gets at half correct

4. Joe gets at least one correct

10 010 1 4 110 5 5 9765625

0 10 10

10

10 1 4 40 5 5 5

0 1010 1 41

0 5 5

5 510 1 45 5 5

Binomial expected value handout

Documents

Chapter 15 Counting and Probability Section 15-1 Counting Problems and Permutations