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0 2 4 6 8 10 WonLost 1234 Year Number of Games Warm-Up 1) In which year(s) did the team lose more games than they won? 2) In which year did the team play the most games? 3) In which year did the team play ten games?

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Math I Day 48 (10-19-09) UNIT QUESTION: How do you use probability to make plans and predict for the future? Standard: MM1D1-3 Today’s Question: How can I find the different ways to arrange the letters in the word “PENCIL”? Standard: MM1D1.b.

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Probability Math I October 19, 2009

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Statistics

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Box Plots

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Control Charts

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Let’s work on some definitions Experiment- is a situation involving chance that leads to results called outcomes. Experiment- is a situation involving chance that leads to results called outcomes. An outcome is the result of a single trial of an experiment An event is one or more outcomes of an experiment. Probability is the measure of how likely an event is. In the previous problem, the experiment is spinning the spinner. In the previous problem, the experiment is spinning the spinner. The possible outcomes are landing on yellow, blue, red or green The possible outcomes are landing on yellow, blue, red or green One event of this experiment is landing on blue. One event of this experiment is landing on blue. The probability of landing on blue is one fourth. The probability of landing on blue is one fourth.

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Probability of an event The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. P(A)=The number of ways an event can occur P(A)=The number of ways an event can occur Total number of possible outcomes Total number of possible outcomes P(blue)= number of ways to land on blue P(blue)= number of ways to land on blue total number of colors total number of colors

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If P = 0, then the event _______ occur. Probability If P = 1, then the event _____ occur. It is ________ It is ______ So probability is always a number between ____ and ____. impossible cannot certain must 1 0

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All of the probabilities must add up to 100% or 1.0 in decimal form. Complements Example: Classroom P (picking a boy) = 0.60 P (picking a girl) = ____ 0.40

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A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. Experiment: A marble chosen at random. Possible outcomes: choosing a red, blue, green or yellow marble. Possible outcomes: choosing a red, blue, green or yellow marble. Probabilities: Probabilities: P(red) = number of ways to choose red = 6 = 3 P(red) = number of ways to choose red = 6 = 3 total number of marbles 22 11 total number of marbles 22 11 P(green)= 5/22, P(blue)= ?, P(yellow)= ?

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There are 3 ways to roll an odd number: 1, 3, 5. You roll a six-sided die whose sides are numbered from 1 through 6. What is the probability of rolling an ODD number? Ex. P 1 2 = 3 6 =

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Theoretical or experimental? We can calculate what our probabilities should be (theoretical values), but that is not always what happens in a real experiment. We could spin the spinner and land on the blue sector every time (experimental values). We can calculate what our probabilities should be (theoretical values), but that is not always what happens in a real experiment. We could spin the spinner and land on the blue sector every time (experimental values). That’s not very likely, but it could happen That’s not very likely, but it could happen

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Favorable outcomes Suppose you have the four color spinner-(red, blue, green and yellow. The probability of spinning a red is ¼, but how many reds should you get if you spin it 20 times? Suppose you have the four color spinner-(red, blue, green and yellow. The probability of spinning a red is ¼, but how many reds should you get if you spin it 20 times? 20 * ¼ = 5 times, you should theoretically land on red 5 times in 20 spins. 20 * ¼ = 5 times, you should theoretically land on red 5 times in 20 spins. Does that always happen with the spinners- why don’t the values always match what you expect? Does that always happen with the spinners- why don’t the values always match what you expect?

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Tree Diagrams Tree diagrams allow us to see all possible outcomes of an event and calculate their probabilities. This tree diagram shows the probabilities of results of flipping three coins. Calculate P (heads), P(2heads,1 tail), P(tails)

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Probability: Permutations

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Use an appropriate method to find the number of outcomes in each of the following situations: 1. Your school cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or orange juice. If you purchase one sandwich, one side item and one drink, how many different lunches can you choose? Sandwich(2)Side Item(2) Drink(3) Outcomes chicken tuna There are 12 possible lunches. chips fruit chips fruit apple juice orange juice milk chicken, chips, apple chicken, chips, orange chicken, chips, milk chicken, fruit, apple chicken, fruit, orange chicken, fruit, milk tuna, chips, apple tuna, chips, orange tuna, chips, milk tuna, fruit, apple tuna, fruit, orange tuna, fruit, milk

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Multiplication Counting Principle At a sporting goods store, skateboards are available in 8 different deck designs. Each deck design is available with 4 different wheel assemblies. How many skateboard choices does the store offer? 32

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Multiplication Counting Principle A father takes his son, Marcus, to Wendy’s for lunch. He tells Marcus he can get a 5 piece nuggets, a spicy chicken sandwich, or a single for the main entrée. For sides, he can get fries, a side salad, potato, or chili. And for drinks, he can get a frosty, coke, sprite, or an orange drink. How many options for meals does Marcus have? 48

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Many mp3 players can vary the order in which songs are played. Your mp3 currently only contains 8 songs. Find the number of orders in which the songs can be played. 1st Song 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Outcomes There are 40,320 possible song orders. In this situation it makes more sense to use the Fundamental Counting Principle. 8 The solutions in examples 3 and 4 involve the product of all the integers from n to one. The product of all positive integers less than or equal to a number is a factorial. 7 6 5 4 3 2 1= 40,320

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Factorial EXAMPLE with Songs ‘eight factorial’ The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial. factorial. 8! = 8 7 6 5 4 3 2 1 = 40,320

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Factorial Simplify each expression. a.4! b.6! c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners. 4 3 2 1 = 24 6 5 4 3 2 1 = 720 = 5! = 5 4 3 2 1 = 120

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5. The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer. President(15) Vice(14) Secretary (13) Treasurer(12) Outcomes There are 32,760 permutations for choosing the class officers. In this situation it makes more sense to use the Fundamental Counting Principle. 151413 12 = 32,760

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Let’s say the student council members’ names were: John, Miranda, Michael, Kim, Pam, Jane, George, Michelle, Sandra, Lisa, Patrick, Randy, Nicole, Jennifer, and Paul. If Michael, Kim, Jane, and George are elected, would the order in which they are chosen matter? President Vice President Secretary Treasurer Although the same individual students are listed in each example above, the listings are not the same. Each listing indicates a different student holding each office. Therefore we must conclude that the order in which they are chosen matters. Is MichaelKim Jane George the same as… Jane Michael George Kim ?

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Permutation Notation

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Permutation When deciding who goes 1 st, 2 nd, etc., order is important. *Note if n = r then n P r = n! A permutation is an arrangement or listing of objects in a specific order. The order of the arrangement is very important!! The notation for a permutation: n P r = n is the total number of objects r is the number of objects selected (wanted)

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Permutations Simplify each expression. a. 12 P 2 b. 10 P 4 c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible? 12 11 = 132 10 9 8 7 = 5,040 = 20 P 4 = 20 19 18 17 = 116,280

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Classwork Practice Workbook Lesson 6.2 - #12-20

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Homework Page 344 #1-6, 25-28

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