1. Counting and Probability
Chapter 15
Counting and Probability

2. Counting Problems and Permutations
Section 15-1
Counting Problems and Permutations

8. The result of a succession of events Definition Event
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
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

9. The fundamental counting principle
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
Total possible number of dinners = 2(4)(2) = 16

15. The set of n objects taken n at a time
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
The set of n objects taken n at a time

16. Find the number of possible arrangements of the set {3,4,7}
Find the number of possible arrangements of the set {a,b,c,d}

22. Possible Codes:
_____ _____ _____ _____
_____ _____ _____
_____ _____
_____ 2 2 2 2 2 2 2 2 2 2

23. 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).

27. HW #15.1 Pg Odd

29. Chapter 15 Counting and Probability
15-2
Permutations For Special Counts

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

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

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

34. 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?

35. 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?

36. 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).

38. Three Types of Permutations

39. 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?

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

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

42. Objective: Find circular permutations

43. 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

45. HW #15.2 Pg , Odd, 20-34

46. 15-3
Combinations

47. 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.

48. Find the value of each expression.

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

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

53. 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?

54. 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?

55. Winning the Lottery
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?

58. A hamburger restaurant advertises "We Fix Hamburgers 256 Ways
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.

60. HW #15.3 Pg

61. 15-4 Binomial theorem

62. Lesson Pascal’s Triangle and Relation to Combinations
Do some expansions
Solve for x
Straight from the book
HW 15.4 Pg

63. HW 15.4 Pg

64. 15-5 Probability Definition Event
Objective: Compute the probability of a simple event.
15-5 Probability
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.

65. Objective: Compute the probability of a simple event.
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.

66. Objective: Compute the probability of a simple event.

69. Objective: Compute the probability of a simple event.

72. Objective: Compute the probability of a simple event.

73. Objective: Compute the probability of a simple event.
The answers are all determined if you know which questions you will answer correctly and which you will answer incorrectly.

74. Objective: Compute the probability of a simple event.

75. Objective: Compute the probability of a simple event.

76. HW #15.5 Pg

77. 15-6 Probability of Compound Events

78. 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.

79. Drawing a Face Card or an Ace are mutually exclusive
Mutually Exclusive Events
Two Events are mutually exclusive if they cannot occur at the same time
Example
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?
Drawing a Face Card or an Ace are mutually exclusive

80. Drawing a Heart or a Face Card are not mutually exclusive
Mutually Exclusive Events
Two Events are mutually exclusive if they cannot occur at the same time
Example
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?
Drawing a Heart or a Face Card are not mutually exclusive

81. If A and B are mutually Exclusive, then P(AB) = 0
Objective: Find the probability that one event or another will occur.
If A and B are mutually Exclusive, then P(AB) = 0

82. Not Mutually Exclusive
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.
Not Mutually Exclusive
Mutually Exclusive
Examples mut/not 1

83. A and B are mutually exclusive
A: Select an ace B: Select a face card.
A and B are mutually exclusive

84. A and B are NOT mutually exclusive
SOLUTION
A: Select a heart B: Select a Face Card
A and B are NOT mutually exclusive

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

86. 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.

87. 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

88. 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.
Independent/Dependent

89. Event A: You roll two dice
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?
Independent
Dependent
Examples Ind/Dep 1

90. 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?
Dependent
Sample Space Method:
Examples Ind/Dep 1 -compare-sample-space

91. 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)
Sample Space Method:
P(king on 1st card  queen on 2nd card) =
Multiplication rule order matters
Probability of a king and queen in that order
Examples Ind/Dep 2 -compare-sample-space

92. SOLUTION
A: Getting more than $500 on the first spin B: Going bankrupt on the second spin.
The two events are independent.

93. 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.

94. HW #15.6a Pg Odd

95. 15-6 Day 2 Probability of Compound Events

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

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

117. Objective: Compute the probability of a simple event.
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.

118. Objective: Compute the probability of a simple event.
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.

119. Objective: Compute the probability of a simple event.
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?

121. HW #15.6b Pg Even, 24, 26-29

122. More Fun With Probability

123. Objective: Compute the probability of a simple event.
Suppose 5 cards are drawn from a deck of 52 cards. What is the probability that you draw 5 Spades?

124. Objective: Compute the probability of a simple event.
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?

125. Objective: Compute the probability of a simple event.
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?

126. Objective: Compute the probability of a simple event.

127. Objective: Compute the probability of a simple event.

128. HW#R-15 Pg

129. 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”.

130. Example
Find the expected value of X, where the values of X and their corresponding probabilities are given by the following table:
SOLUTION

131. Two numbers can be selected from the five numbers in C(5, 2) = 10 ways
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.
Two numbers can be selected from the five numbers in C(5, 2) = 10 ways
Score Pairs
5 (1, 4), (2, 3)
1 (1, 2),(1, 3), (1, 5)
2 (2, 4), (2, 5)
3 (3, 4), (3, 5)
4 (4, 5)
E(X) = 1(3/10) + 2(2/10) + 3(2/10) + 4(1/10) + 5(2/10) = 2.7

133. Expected Value Roulette
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 $1 bet on a single number is, considering all 38 possible outcomes:                                                                                                               
which is about −$ Therefore one expects, on average, to lose over five cents for every dollar bet.

134. Expected Value Gain x Sample Point Probability $290 Customer lives
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 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 $( ,000).

135. Expected Value Gain x Sample Point Probability $290 Customer lives
.999
$290-$10,000
Customer dies
.001
The insurance company expects to make a $280 profit on the deal at the end of the first year.

136. Expected Value Gain x Sample Point Probability 7,000,000 Win
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 /

137. 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.
What is the probability he will win
What is the probability he will loose?
What is the expected value of the game?

138. Binomial Probability Model
On 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?

139. 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)

140. 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

141. A fair quarter is flipped three times
A fair quarter is flipped three times. Find the following probabilities:
You get exactly three heads
You get exactly two heads
You get two or more heads
You get exactly three tails

142. 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
Joe gets all 10 correct
Joe gets all 10 wrong
Joe gets at half correct
Joe gets at least one correct

143. Binomial expected value handout