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UNIT 4: Applications of Probability UNIT QUESTION: How do you calculate the probability of an event? Today’s Question: What is a permutation or combination and how do we use it to solve problems?

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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.

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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

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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)= ?, 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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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.

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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 Tanner to Wendy’s for lunch. He tells Tanner he can get the 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 milk, coke, sprite, or the orange drink. How many options for meals does Tanner have? 48

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Your iPod players can vary the order in which songs are played. Your iPod currently only contains 8 songs (if you’re a loser). 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 solution in this example involves the product of all the integers from n to one (n is representing the starting value). 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 Vice Secretary Treasurer 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: Hunter, Bethany, Justin, Madison, Kelsey, Mimi, Taylor, Grace, Maighan, Tori, Alex, Paul, Whitney, Randi, and Dalton. If Hunter, Maighan, Whitney, and Alex 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 HunterMaighan Whitney Alex the same as… Whitney Hunter Alex Maighan?

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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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Permutation Notation

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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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24 1.Bugs Bunny, King Tut, Mickey Mouse and Daffy Duck are going to the movies (they are best friends). How many different ways can they sit in seats A, B, C, and D below? 2. Coach is picking a captain and co-captain from 15 seniors. How many possibilities does he have if they are all equally likely? ABCD 210

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Combinations A selection of objects in which order is not important. Example – 8 people pair up to do an assignment. How many different pairs are there?

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Combinations ABACADAEAFAGAH BABCBDBEBFBGBH CACBCDCECFCGCH DADBDCDEDFDGDH EAEBECEDEFEGEH FAFBFCFDFEFGFH GAGBGCGDGEGFGH HAHBHCHDHEHFHG

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The number of r-combinations of a set with n elements, where n is a positive integer and r is an integer with 0 <= r <= n, i.e. the number of combinations of r objects from n unlike objects is Combinations

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Example 1 How many different ways are there to select two class representatives from a class of 20 students?

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Solution The answer is given by the number of 2-combinations of a set with 20 elements. The number of such combinations is

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Example 2 From a class of 24, the teacher is randomly selecting 3 to help Mr. Griggers with a project. How many combinations are possible?

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Your turn! For your school pictures, you can choose 4 backgrounds from a list of 10. How many combinations of backdrops are possible?

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Clarification on Combinations and Permutations "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

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Clarification on Combinations and Permutations "The combination to the safe is 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.

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To sum it up… If the order doesn't matter, then it is a Combination. If the order does matter, then it is a Permutation. A Permutation is an ordered Combination.

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