Algebra-2 Counting and Probability

Quiz 10-1, Which of these are an example of a “descrete” set of data? 2.Make a “tree diagram” showing all the ways the letters ‘x’, ‘y’, and ‘z’ can be arranged in order. ‘x’, ‘y’, and ‘z’ can be arranged in order. 3. You are paying for groceries at the store. You have the following bills: $100, $50, $20, $10, $5, $2, and $1. What are number of different sums of money that you can pull out of your wallet if you pull out 3 bills without looking?

Counting How many ways can you arrange the letters ‘x’, ‘y’, and ‘z’ in order? ___ ___ ___ 321 Why isn’t it ? ___ ___ ___ 333 You can’t re-use a letter in this situation. Can you think of a situation where you could re-use a number or a letter? Phone numbers Social security numbers License plates Go to demo

Vocabulary “arranging without replacement: when you use an item in the arrangement, it is “used up” and can’t be used again. the arrangement, it is “used up” and can’t be used again. “arranging with replacement: when an item is used in one position in an arrangement, it can be used again in another position in an arrangement, it can be used again in another position in the arrangement. position in the arrangement. Think of arranging people in a line. Once a person is in the front of the line, he cannot also be in the back of the line front of the line, he cannot also be in the back of the line at the same time. at the same time. Think of arranging numbers and Letters on a license plate: the previous number or letter can be used again.

Effect on Muliplication Principle of counting ( Product of the # of options for each step) arranging without replacement: arranging with replacement: Arranging 3 numbers on a licence plate. Arranging 3 people in a line. Factorial

Your turn: Which is it (with or without replacement) for: 1. Assigning 3 committee members to the positions of: “Pres”, “Vice-Pres”, and “Secretary” “Pres”, “Vice-Pres”, and “Secretary” 2. The total number of social security numbers with 9 digits.

Using the Multiplication Principle If a license plate has three letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. L L L # # # How many possibilities for the 1 st position (letter)? 26

Using the Multiplication Principle L L L # # # How many possibilities for the 2 nd position? 26 * 26

Using the Multiplication Principle L L L # # # How many possibilities for the 3 rd position? 26 * 26

Using the Multiplication Principle L L L # # # How many possibilities for the 4 th position (number)? 26 * 26 * 10

Using the Multiplication Principle L L L # # # How many possibilities for the 5 th position? 26 * 26 * 10

Using the Multiplication Principle L L L # # # How many possibilities for the 6 th position? 26 * 26 * 10 Total number of distinct license plates = 676,000

Your Turn: 3. How many distinct license plates can be made using 6 digits (numerals 0 – 9)? (# # # # # #) 6 digits (numerals 0 – 9)? (# # # # # #) 4. How many distinct license plates can be made using 2 digits (numerals 0 – 9) and 4 letters ( A – Z) ? 2 digits (numerals 0 – 9) and 4 letters ( A – Z) ? (# # L L L L) (# # L L L L) 1,000,000 45,697,600

Your Turn: Count the number of different 8-letter “words” (groups of 8 letters) that can be formed using the letters in the word COMPUTER. Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations. 5.

What if two of the letters are the same? Count the number of different 4-letter “words” that can be formed using the letters in the word “WAAG”. “WAAG”. Let “A” be the 1 st A. Let “A” be the 2 nd A. What’s the difference between AAWG and AAWG? There’s no difference!! They are not distinguishable from each other. So we really have “double counted” from each other. So we really have “double counted” a bunch of words. WAAG ”.

What if two of the letters are the same? Count the number of different 4-letter “words” that can be formed using the letters in the word “WAAG”. “WAAG”. AAWG (AAWG) is one example of double counting. To remove the “double counting” we must divide out the number of possible ways to divide out the number of possible ways to permutate A and A permutate A and A AWAG (AWAG) is another example of double counting. We must divide by 2!.

What if three of the letters are the same? Count the number of different 5-letter “words” that can be formed using the letters in the word “WAAAG”. “WAAAG”. AAAWG AAAWG AAAWG AAAWG AAAWG AAAWG To remove the “double counting” we must divide out the number of ways to permutate A, A and A divide out the number of ways to permutate A, A and A These are all examples of the same word and have been “double counted”. “double counted”. We must divide by 3! AAAWG AAAWG AAAWG AAAWG AAAWG AAAWG WAAAG” “ WAAAG”.

Distinguishable Permutations We must “divide out” the permutations of the same object that result in indistinguishable arrangements. that result in indistinguishable arrangements. If a set 12 items to be permutated has 3 objects of one kind, and 4 objects of another kind, and 5 objects of another kind, and 4 objects of another kind, and 5 objects of another kind, then the number of distinguishable ways to arrange the 12 then the number of distinguishable ways to arrange the 12 items is: items is:

Distinguishable Permutations In general, we find the number of distinguishable permutations when using some elements that are indistinguishable when using some elements that are indistinguishable as follows: as follows: If a set N items to be permutated has A objects of one kind, and B objects of another kind, and C objects of another kind, and B objects of another kind, and C objects of another kind, and A + B + C = N then the number of distinguishable ways and A + B + C = N then the number of distinguishable ways to arrange the N items is: to arrange the N items is:

Your Turn: You have the following bills in your wallet: three $20’s, four $10’s, five $5’s, and six $1’s What is the number of distinct ways you could pay out the bills one at a time? 6.

Counting How many 5 card hands are there with all face cards (king, queen, jack). Which is it? This tells you the hands all have 5 face cards. So how many arrangements are there when taking 12 cards and many arrangements are there when taking 12 cards and picking 5 ? picking 5 ? Permutation: (different order  counted separately) Combination: (different order  not counted separately)

Your Turn: How many 5 card hands are there with no face cards? 7.

Counting Sometimes there are more than one condition that must be met. How many 5 card hands have all 5 cards the same suite (hearts, diamonds, spades, clubs). (hearts, diamonds, spades, clubs). Combination: (different order  not counted separately) 1 st we must pick the suite: 2 nd we must pick the 5 cards from that suite: By the multipication principle: total number hands is:

Counting How many 5 card hands have exactly 2 aces ? Combination: (different order  not counted separately) 1 st we must pick the 2 aces: 2 nd we must pick the other 3 cards: (if the hand has exactly 2 aces, then we must not include the other two aces as possible picks) By the multipication principle: total number of hands is:

Your Turn: How many 5 card hands are there with two fives and two sixes? 8. Hint: (1) pick the 2 fives, (2) pick the 2 sixes, (3) Pick the last card. Use the multiplication rule.

Probability “What’s the chance of something happening?” “There is a 100% chance it will rain today.” “There is less than a 5% chance you will be picked.

Your turn: Can probability be equal to 50%? What is the largest number that a probability can be? What is the smallest number that a probability can be? Can there be a (– 20)% chance something will happen?

Probability When discussing probability, you can use either “%”, fraction, or the decimal equivalent.  “There is a 40% chance of thunderstorms today.” In mathematics, we convert % to the decimal equivalent or leave it in fraction form.  “The probability of rain today is 0.4.”

Theoretical Probability The probability of an event occurring: There are 4 different colored marbles in a bag (red, blue, green and yellow). What is the probability of pulling out a red one on the first try?

Examples The probability of rolling a ‘5’ using one die. The probability of drawing a “king” from a deck of cards.

The challenge you have is counting the ways to achieve the event (sometimes called successful events) and then counting the total possible outcomes. What is the probility of pulling an A, followed by a B, and then a C out of a bag with the letters ‘A’, ‘B’, and ‘C’ in it ?

Your Turn: 13. What is the probability of picking the correct number when someone asks you to pick a number from 1 to 10. when someone asks you to pick a number from 1 to There are 2 red marbles and 3 green ones in a bag. What is the probability of picking out a red marble on What is the probability of picking out a red marble on the first try? the first try? Probability only works if the events are completely random. Picking a committee using numbers out of a hat or a similar Picking a committee using numbers out of a hat or a similar random method of picking them is the only way that random method of picking them is the only way that probability will work. probability will work.

Your Turn: people are trying to be selected for a 3 person committee. You don’t know any of the people. What is the probality of you guessing who committee. You don’t know any of the people. What is the probality of you guessing who will be on the committee? will be on the committee? 16. What is the probability of having a 5 card hand with a single pair of kings in it?

Geometric Probability: ratio of areas Assumming that at an arrow randomly hits anywhere in the four square area, what is the probability of hitting in the #1 square? Since all squares have the same area, and #1 is ¼ of the total area  probability is ¼. and #1 is ¼ of the total area  probability is ¼.

Geometric Probability : the area of each ring is given. If an arrow will randomly hit anywhere inside of the red circle, what is the probability of hitting the center blue circle? circle, what is the probability of hitting the center blue circle?

Geometric Probability 17. What is the probability of hitting the pink ring? 18. What is the probability of hitting either the pink or dark blue ring?

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Probability using combinations and permutations. At the Roy High School Talent show 7 musicians are scheduled to perform. What is the probability that they will perform in alphabetical order of their last names (nobody has the same last name) ? There is only one order of performers that is in alphabetical order. How many ways can you arrange 7 persons names in order? Is this a permutation or combination?

Probability using combinations and permutations. At the Roy High School Talent show 7 musicians are scheduled to perform. 3 performers are girls and 4 are boys. What is the probability that all 3 girls will be first? How many ways can you get the first 3 performers to be girls? How many ways can you arrange 3 of 7 people in order? Is this a permutation or combination?

Your Turn: 19. At the Roy High School Talent show 7 musicians are scheduled to perform. They are: Bill, Brad, Bob, and Brody (boys) and Kylee, Kaylee, and Kyla (3 girls). What is the probability that 2 boys will be first?

Your Turn: 20. What is the probability of getting 4 aces in a randomly dealt hand of 4 cards?

Your Turn: 21. A lottery uses numbers 1 thru numbers are drawn randomly. The order in which you choose the numbers doesn’t matter. What is the probability of winning the lottery if you buy one ticket (assume nobody else picks the winning number) ? How many ways can you get the 6 out of 6 correct numbers? How many ways can you pick 6 of 46 numbers? Is picking 6 of 46 a permutation or a combination?

Cards: What is the probability of getting 4 aces a randomly dealt hand of 5 cards?

Your turn: 17. What is the probability of getting 3 aces and 2 kings from randomly dealt hand of 5 cards?