Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different yogurts can be ordered? Multiplication Principle: Suppose a task T 1 can be performed in N 1 ways, a task T 2 can be performed in N 2 ways,..., and a task T n can be performed in N n ways. Then, the number of ways of performing the tasks T 1, T 2,..., T n in succession is given by the product: N 1 N 2 N n Example 2: A coin is tossed 3 times, and the sequence of heads and tails is recorded. Determine the number of outcomes. Example 3: An auto manufacturer has 3 different subcompact cars in the line. Customers selecting one of these cars have a choice of 3 engine sizes, 4 body styles, and 3 color schemes. How many different selections can a customer make? 1
Example 4: How many three-letter words that have exactly one vowel can be made using the first seven letters of the alphabet where using a letter twice is permitted but having two consonants next to each other is not? Example 5: How many five-digit numbers can be formed if a) Zero is not the first digit? b) Zero cannot be the first digit and no digit can be repeated? c) Zero cannot be the first digit, no digit can be repeated, and each number formed must be even? 2
Example 6: Jack and Jill and 5 of their friends go to the movies. They all sit next to each other in the same row. How many ways can this be done if a) there are no restrictions? b) Jill must sit in the middle? c) Jill sits on one end of the row and Jack sits on the other end of the row? d) Jack, Jill, or John sit in the middle seat? e) Jack, Jill, and John sit in the middle seats? f) Jack and Jill must sit next to each other? g) Jill must not sit next to Jack? 3
Example 7: Suppose we want to seat 12 people in a row of 12 seats. How many arrangements are possible? The above product is called a factorial: n! = n(n 1)(n 2) 3 2 1 Note: 0! = 1 Example 8: How many ways can we select 5 people from a group of 12 and arrange them in 5 chairs? Definition: If we have n distinct elements and we want to take r of them in an arrangement, we say that the number of arrangements of n things taken r at a time is: n! (n r)! Example 9: How many ways can we select 25 people from a group of 35 and arrange them in 25 chairs? 4
Arrangement of n objects, not all distinct: Given a set of n objects in which n 1 are alike of one kind, n 2 are alike of another,..., n r alike of another so that n 1 + n 2 + + n r = n then the number of arrangements of the n objects taken n at a time is: n! n 1!n 2! n r! Example 10: Suppose we have 2 red marbles, 3 green marbles, and 1 blue marble. If we want to line the marbles up in a row, how many distinguishable arrangements of the 6 marbles are there? Example 11: How many distinguishable arrangements can we make from the letters in the word Mississippi? Section 2.1 Homework Problems: 13, 17, 21, 23, 25, 29, 31, 33, 35, 41 and Counting Handout 1 5
Section 2.2 - Combinations Definition: The number of combinations of n items taken r at a time is: C(n,r) = n! (n r)!r! Example 1: Example 2: Suppose a high school choir made of 11 students decides to send 2 members to a duet competition. a) How many pairs are possible? b) If it is decided that one particular member is to go, how many different pairs are possible? c) If there are 8 girls and 3 boys in the choir, how many pairs will include at least one boy? 6
Example 3: Suppose we have a bag containing 6 purple, 3 red, and 7 green candies. You choose 5 pieces at random. a) How many samples of 5 candies can be chosen? b) How many samples are there in which all the candies are green? c) How many samples are there in which they are all red? d) How many samples are there in which there are 2 purple and 1 red? e) How many samples are there in which there are no purple candies? f) How many samples contain at least 1 purple? g) How many samples contain exactly 2 purple or exactly 2 green candies? 7
Example 4: Suppose we are playing the lottery in which we must choose 6 from 50 numbers. a) How many different lottery picks could we choose if the order we choose our numbers in does not matter? b) How many ways are there to choose no winning numbers? c) How many ways are there to choose at least 3 winning numbers? Example 5: In how many ways can a committee be formed with a chair, a secretary, a treasurer, and four additional people if they are to all be chosen from a group of ten people? 8
Example 6: An investor has selected a mutual fund to invest his money in. He plans on observing its performance over the next ten years. He will consider the year a success (S) if the mutal fund performs above average and a failure (F) otherwise. a) How many different outcomes are possible? b) How many different outcomes have exactly six successes? c) How many different outocmes have at least three successes? Example 7: Five cards are randomly selected from a standard deck of 52 cards to form a poker hand. Determine the number of ways a person can be dealt a full house (that is a three of a kind and a two of a kind) Section 2.2 Homework Problems: 1, 15, 17, 21, 25, 29, 31, 33-43 (odd) and Counting Handouts 2 and 3 9
Section 2.3 - Probability Applications of Counting Principles Computing the probability of an event in a uniform sample space: Let S be a uniform sample space and let E be any event. Then, P(E) = n(e) n(s) where n(e) is the number of outcomes in E and n(s) is the number of outcomes in S. Example 1: Four marbles are selected at random without replacement from a bowl containing five white and eight green marbles. Find the probability that at least two of the marbles are white. Example 2: An unbiased coin is tossed six times. What is the probability that the coin will land heads a) Exactly three times? b) At most three times? c) On the first and the last toss? 10
Example 3: a) An exam consists of ten true-or-false questions. If a student randomly guesses on each question, what is the probability that he or she will answer exactly six questions correctly? b) An exam consists of ten mutliple choice questions each having five choices of which only one is correct. If a student randomly guesses on each question, what is the probability that he or she will answer exactly six questions correctly? Example 4: Two cards are selected at random without replacement from a well-shuffled deck of 52 playing cards. Find the probability that two cards of the same suit are drawn. 11
Example 5: Thirty people are selected at random. a) What is the probability that none of the people in this group have the same birthday? b) What is the probability that at least two people in this group have the same birthday? Section 2.3 Homework Problems: 1, 5, 9-29(odd) 12
Section 2.4 - Bernoulli Trials Definition: A bernoulli trial is an experiment in which the outcome can be either of two possible outcomes: success and failure. Definition: A binomial experiment (OR bernoulli process OR repeated bernoulli trial) is just a repetition of many bernouilli trials. A binomial experiment has the following properties: 1. The number of trials in the experiment is fixed. 2. There are 2 possible outcomes in each trial: success and failure 3. The probability of success in each trial is the same. 4. Trials are independent of each other. Example 1: Suppose a basketball player makes on average two free throws out of every three attempted and that success and failure on any one free throw does not depend on the outcomes of the other shots. If the player shoots ten free throws, find the probability of making exactly six of them. Calculating a Binomial Probability: 1. Determine if the experiment is binomial. 2. Determine the number of trials (n). 3. Define success in the experiment and determine the probability of that success occuring (p). 4. Determine the number of successes desired (r). 5. Calculate the desired probability: (a) By hand, the probability is found by doing the following calculation: (b) We can use the calculator functions binompdf( or binomcdf( to do these calculations for us. Example 2: A fair die is cast 4 times. Compute the probability of obtaining exactly one 6 in the four throws. 13
Example 3: A biology quiz consists of 8 multiple choice questions. Each question has five choices of which only one is correct. If a student randomly guesses on each question, what is the probability of a) getting at most 3 questions right? b) getting at least 2 questions right? c) getting more than 2 but less than 6 questions right? Example 4: Studies have shown that 30% of babies are delivered by C-section. If 95 newborns are randomly selected, what is the probability that at least 18 but fewer than 35 of them were delivered by C-section? Section 2.4 Homework Problems: 1, 7, 13, 15, 17, 21-25 (odd), 29, 35, 39, 43 14