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Finite Math B: Chapter 8 Counting Principles, Further Probability Topics

Chapter 8: Counting Principles; Further Probability Topics

8.1 The Multiplication Principle; Permutations

Recall from Chapter 7 and Algebra that the probability of event E in a sample space of equally likely outcomes S is defined as:

In order to find probability, we need to be able to ______________the number of elements in the set that represents our event and our sample space. This can be quite horrible to do sometimes, so for efficiency, we will spend some time learning counting shortcuts.

The Multiplication Principle

Very good for selecting 1 item from each of several different groups or for making several different choices.

Example 1: A combination lock can be set to open to any 3 digit sequence. How many different combinations are possible?

“digits” = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 = _______

________ x ________ x ________ 1st 2nd 3rd

Example 2: How many three-letter sequences are Example 3: How many different ways can you arrange 5 possible if the letters are not allowed to repeat? books on a shelf.

Example 4: The Local “House of Noodles” has aspecial where you can choose any of 4 different pastasand 5 different sauces. How many pasta bowls with 1type of pasta and 1 type of sauce are possible?

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Probability of event E

P (E )=n(E)n(S)


Be Careful about “At least” or “At Most” type ProblemsExample 5: A artist has 5 different paintings. He wants to arrange at most 3 paintings on the wall. How many possible arrangements are there?

FactorialsMultiplication principle often leads to patterns like:

5 x 4 x 3 x 2 x 1

This is easier calculated as a factorial:

5! = _______

Example 1: How many ways can I design a Example 2: Nine runners are in a race. How many finishplaylist of 7 songs if I don’t want to repeat any orders are possible? Assume ties are not allowed.song on the list?

Permutations

A permutation is a special type of counting principle problem where you are arranging/ordering objects from a set, but you don’t necessary plan to use them all.

To count permutatations you can use:

1. Counting principle 2. Formula 3. nPr command on calculator

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Example 1: In mid 2007, eight candidates sought the Democratic nomination for president. In how many ways could voters rank their first, second, and third choices?

Counting Principle Formula Calculator

Example 2: How many batting lineups (order is important) Example 3: You pack 10 shirts in your bag for aof 9 players can you make from a team of 15? trip. In how many different ways can you wear

the shirts during the first week of your vacation?(7 days, 1 shirt per day)

Permutation with repetition

Example 1: In how many ways can you arrange the Example 2: In how many ways can you arrange letters of the word “STAR” ? the letters of the word “MOON”?

Example 3: How many distinguishable permutations are there of the word “MISSISSIPPI”?

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Permutation with repetition:When arranging a set of n objects, where object 1 repeats R1 times, object 2 repeats R2 times, etc., the number of DISTINGUISHABLE PERMUTATIONS of the n objects can be found by:

n !R1!×R2 !×…×Rn !


Counting Potpourri

Example 1: A vendor is selling red and blue balloons. In how many ways can he hand out 4 red and 7 blue balloons to the next 11 children?

Example 2: (#23 pg 439) A child has a set of differently shaped plastic objects. There are 3 pyramids, 4 cubes, and 7 spheres.

a) In how many ways can she arrange the objects in a row if each is a different color?

b) How many arrangements are possible if objects of the same shape must be grouped together AND each object is a different color?

c) In how many ways distinguishable ways can the objects be arranged in a row if objects of the same shape are also the same color, but need not be grouped together?

d) In how many ways can you select 3 objects, one of each shape, if the order in which the objects are selected does not matter and each object is a different color?

e) In how many ways can you select 3 objects, one of each shape, if the order in which the objects are selected matters and each object is a different color?

Example 3: Your favorite band is performing a short concert with 4 new songs and 5 old songs.

a) How many possible song orders are there? b) How many song orders are there if an old song will be played first?

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8.2 Combinations

Counting Principle.Good for – choosing 1 thing from each of several different sets.Quickly counting possibilities when multiple choices are involved.

Example:A bank PIN is created by choosing two letters followed by two digits. How many codes are possible if letters and numbers are allowed to repeat?

Permutations.Used for – choosing r things out of a pile of n things where order is significant

Example: How many ways can you arrange 5 paintings on a wall?

Example: How many ways can 50 runners come in 1st, 2nd, & 3rd in a race if no ties are allowed?

-------------------------------------------------------------------------------------------------------------------------------------------------------“Rank your three favorite movie genres: comedy, action,sci-fi, horror, romance, western documentary.”

Chip: {sci-fi, horror, action}

Dale: {horror, sci-fi, action}

“Choose your three favorite movie genres: comedy, action, sci-fi, horror, romance, western, documentary”

Chip: {sci-fi, horror, action}

Dale: {horror, sci-fi, action}

A subset listed without regard to order is called a Combination.

Other common notation:

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horrorsci-fiaction

sci-fihorroraction

Me too!

I picked sci-fi, horror, and


Example 1: Example 2:How many committees of 3 people can be formed A salesman has 10 accounts in a certain city. Infrom a group of 8 people? how many ways can he choose 2 to visit today?

BE CAUTIOUS ABOUT “AT LEAST” and “AT MOST” STATEMENTS

Example 3: A manager can recommend at most 4 of his 20 employees for promotion. In how many ways can he submit the names for promotion?

Example 4: Cupcake Paradise has 5 new flavors of cupcakes. You decide you want to try at least 3 of the new flavors. How many orderings would allow you to try at least 3 new flavors?

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Example 5: A teacher is recommending four students for student council. There are 10 students eligible.

a) How many ways can four students be chosen from the 10?

b) How many ways could four students from the 10 be chosen for leadership positions? (pres/vice/secret./treas.)

Example 6: Five cards are dealt from a standard 52 card deck.

a) How many five card hands are possible?

b) How many five card hands are possible that contain only hearts?

“Combinatorics” is using combinations and the counting principle to count events made up of multiple parts.c) How many ways are there to get a “Dead Man’s Hand” of aces over eights? (3 aces and 2 eights)

d) How many ways are there to get a five card had with exactly 3 hearts?

e) How many five card hands have exactly 4 face cards?

Extra Super Sneakyf) How many ways can you get a “full house”? (3 of one card, 2 of another)

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Example 6: Snow White has a basket of apples. Each apple looks a little different. Seven apples are perfectly fine. Three have been poisoned.

a) In how many ways can she select eight apples?

b) In how many ways could she choose 3 apples, none of which are poisoned?

c) In how many ways could she choose 5 apples, exactly 2 of which have been poisoned?

Example 7: You are at Dunkin’ Donuts. You ask for a random selection of a dozen donuts. You end up with 12 unique donuts: 3 cream filled, 4 iced, 2 cake, and 3 yeast.

a) You drop the box. Five donuts fall sadly into the dirt. The rest are OK. In how many ways could five donuts be damaged?

b) In how many ways could you choose 3 donuts to give to friends if one of the donuts MUST be cake?

c) In how many ways could you line up 5 donuts in a row to take a photo for your new Tumblr “Five-of-Stuff”?

REVIEW:Counting Principle Permutation Combination

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8.3 Probability Applications of Counting Principles

Many probabilities are difficult to calculate due to the large number of possibilities. We’ll be using Permutations and Combinations to help us out.

Recall: E is a subset of a sample space S of equally likely outcomes.

Example 1: The Environmental Protection Agency is considering inspecting 6 plants for environmental compliance: 3 in Chicago, 2 in Los Angeles, and 1 in New York. Due to a lack of inspectors, they decide to inspect 2 plants selected at random. Each plant is equally likely to be selected, but no plant will be selected twice.

This is a COMBINATION/COMBINATORICS problem because we are selecting ____ of _____ with no regard for the order.

a) In how many ways can two plants be selected?

b)What is the probability that both plants will be c) What is the probability that 1 Chicago Plant in Chicago? and 1 Los Angeles plant will be selected?

Example 2: From a group of 22 nurses, 4 are to be selected at random to present a list of grievances to management. One of the nurses is Madame Pomfrey, who specializes in magical cures.


a) In how many ways can this be done?

b) Find the probability that Madame Pomfrey c) What is the probability that Madame Pomfrey will be among the 4 selected. is NOT selected?

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Probability of event E:

P (E )= n(E)n(S)


Example 3: In the card game Poker, a hand of five cards is dealt to each player from a deck of 52 cards.


a) A “heart flush” is a hand containing only hearts. What is the probability of getting a heart flush?

b) A “full house” is a hand containing three of one card and two of a different card. What is the probability of getting a full house with three kings and two queens?

c) What is the probability of getting a hand with four aces?

d) What is the probability of getting a hand with at least one king?

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Example 4: There are 4 diet sodas, 6 regular sodas, and 3 cans of lemonade in the refrigerator. You yell at your little brother and tell him to bring you 3 cans of soda.


a) What is the probability that he will bring you one of each type of drink?

b) What is the probability that all three cans will be the same type of drink?

Example 5: Amazon Instant Videos currently has several movies on their homescreen you can buy and download to your Kindle Fire to take and watch on vacation. There are 5 action, 4 romance, 3 comedy, and 2 documentaries being featured.


a) How many ways can you choose 4 movies to buy and download from the featured films?

b) What is the probability of choosing all action c) What is the probability of choosing 2 comedy movies? and 2 action movies?

d) What is the probability of choosing more romance than any other type?

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Example 6: You roll a die 4 times. You roll the die, record the number, then roll again. Find the following probabilities.

This is NOT a COMBINATION/COMBINATORICS problem because we are not selecting r items from a set of n objects. Instead, we will need to use the MULTIPLICATION RULE to determine these probabilities.

a) You roll the sequence: 2, 2, 4, 6 b) Your first roll is a “5”

c) You roll all different numbers d) Your sequence does not contain an even number.

Example 7: All the letters in the alphabet are written on cards. You draw four cards and place them in a random order.

This is NOT a COMBINATIONS/COMBINATORICS problem. Although we are choosing _____ of ______, we must consider the ORDER of the objects. Instead, we will need to use the MULTIPLICATION RULE (or PERMUTATIONS) to determine this probability.

a) What is the probability that cards spell out the word: MATH

b) What is the probability that the cards contain no vowels: A, E, I , O, U ?

c) What is the probability that the cards are NOT in alphabetical order?

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8.5 Probability Distributions and Expected Value

Expected value:

Basically, we want to multiply each value by its probability and find the sum of all of these products.

Example:A local symphony decides to raise money by raffling a microwave oven worth $400, a dinner for two worth $80, and 2 books worth $20 each. A total of 2000 tickets are sold at $1 each. Find the expected payback for a person who buys 1 raffle ticket.

Example: A friendly wager is made. Each day Amy and Rory toss a quarter to decide who buys coffee. If the person who “calls it” is correct, the other person buys coffee. If they are wrong, they buy the coffee.

A cup of coffee is $1.20. What are the expected winnings?

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Example: Find the expected value of the probability distribution:

a)

b) A teacher gives a quiz with 8 questions. She records the number of correct answer each student gets and makes the following histogram. What is the expected value? What does the expected value represent?

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2 3 4 5 6 7 8

Number of Correct Responses

Frequency

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€¦ · Web viewRecall from Chapter 7 and Algebra that the probability of event E in a sample space ... letters of the word “STAR” ?the letters ... Counting Potpourri